Example A 76.2 kg rocket is launched vertically upward. The engine supplies a constant thrust of 765 N until it is cut off when the rocket reaches a speed of 233 km/h. What is the altitude of the rocket in kilometers when the engine is cut off? Neglect the effect of air friction. The forces exerted on the rocket are shown in Fig. 4.7b.
SOLUTION: Applying Newton's second law we have
F = ma
Fthrust - mg = ma
765 N - 76.2kg 9.8 m/s2 = 76.2 kg a.
from which we get for the acceleration a = 0.239 m/s2. We obtain a expression for the distance d traveled altitude from Eqs. 2.4 and 2.8.
d = v2 4.6
2a
a is rocket b is thrust mg pointing toward launch pad
and F on top for force thrust.
Rocket Propulsion.
Understanding The Universe
Exercises
Chapter 1
1. Assume that Eratosthene's measurements of the distance from Alexandria to Syene and the angle of the sun at Alexandria at noon on the day of the solstice were accurate. Use these data to determine the length in meters of the stadium he used to express the size of the earth. For the actual size of the earth see Appendix B.
radius of the earth 6,370 km.
mass of the earth 5.98 x 10 24 kg.
2. What was the main defect in Copernicus's model of the heavens? Explain clearly why this was a problem.
3. Mars, as seen from the earth, varies greatly in brightness over time intervals of the order of months. Use drawings to explain clearly why this is so.
4. As seen from the earth Venus is brightest at its crescent phase. The moon is not brightest at its crescent phase. Use words and drawings to explain clearly the difference.
5. From the earth Mercury and Venus are only seen just after sunset or just before sunrise. Use words and drawings to explain clearly why this is so.
Venus rises in the eastern sky after sunset and sets in the western sky just before sunrise.
6. The planet Neptune requires about 165 years to make a complete orbit
around the sun. Use Kepler's third law to estimate Neptune's average
distance from the sun.
Answer 4.5 x 10 9 km.
10 9
4.5
7. Suppose an object is discovered to move in a circular orbit around the sun and that it makes one complete orbit in 583 years. How far is this object from the sun?
Answer 69.8 AU.
8. On New Year's day in 1801 a Sicilian astronomer discovered a faint celestial object that appeared to move relative to the fixed stars. Later, he gave this object the name Ceres, after the Roman agricultural goddess. Before the end of that year, the famous German mathematician Karl Friedrich Gauss established that the object was in a closed orbit around the sun at a mean distance of 2.77 AU. How long does it take Ceres to make a complete orbit around the sun?
9. Suppose a new planet is discovered to be moving around the sun in an elliptical orbit with a semimajor axis 1.16 x 10 10 km. Estimate the orbital period of this planet.
Answer 680 years.
10. Because of a certain peculiarity observed in Mercury's orbit, astronomers in the last century speculated that an undiscovered planet orbited the sun inside the orbit of Mercury. The anomaly in Mercury's orbit was attributed to this planet, which would be close to the sun and difficult to see from the earth. Because of its proximity to the fiery sun this putative planet was called Vulcan. Suppose Vulcan moves in a circular orbit of radius 0.22 AU.
Estimate the time that it would take Vulcan to make a complete orbit around the sun.
Answer 38 days.
11. A minor planet asteroid travels in a circular orbit around the sun making a complete orbit in 5.59 years. Use Kepler's third law to estimate how close this object comes to the earth. What is its maximum distance from the earth? A drawing will be helpful.
Answer 3.2 x 10 8 km 6.2 x 10 8 km.
12. Some scientists have suggested that mass extinctions of life on earth such as the one that killed off the dinosaurs 65 million years ago occur periodically at intervals of 26 million years. One interesting albeit controversial theory that has been proposed to account for these periodic catastrophes calls for a very small dim companion star orbiting around the sun. This star will periodically disturb a distant cloud of matter that surrounds the sun and send a hail of comets hurtling through the inner solar system. At such times the comets would be so numerous that a collision with the earth would be highly probable. Estimate the length of the major axis of the orbit
of this object.
Answer 175,000 AU.
13. In 1977 an object called Chiron with a diameter of about 200 km was
discovered in an elliptical orbit around the sun. At its closest approach to the sun perihelion Chiron is 8.49 AU from the sun. At its greatest distance from the sun aphelion it is 18.91 AU away. From these data calculate the orbital period of Chiron. Explain clearly your reasoning. A drawing showing Chiron's orbit around the sun will help here.
Answer 50.7 years.
14. It is not possible to design an accurate calendar? Why not?
Also the calendar was only created by humans for humans since the calendae has no effect on other species.
15. Explain clearly and completely why the moon shows phases.
16. It takes a little over 27 days for the moon to complete an orbit around the earth but over 29 days are required for the moon to pass through a complete set of phases. By means of words and drawings use the heliocentric suncentered view of the universe to give a clear and complete explanation of this difference.
17. Use words and drawings to explain clearly and completely why we experience seasons on the earth.
18. What is the ecliptic? Why is it so named?
19. Explain clearly and completely how a solar eclipse is produced. Make a drawing showing all of the relevant objects. Under what circumstances do solar eclipses occur? Why do solar eclipses compared to lunar eclipses occur relatively rarely at a given point on the earth's surface?
20. Make a qualitatively correct drawing similar to Fig. 1.11 to illustrate how a total lunar eclipse is produced. Based on your drawing explain clearly why a lunar eclipse may be seen by an observer anywhere on the nighttime side of the earth. Explain clearly why lunar eclipses occur more frequently than solar eclipses.
Exercises
Chapter 2
t As
s cm
1.57 210.0
1.45 180.0
1.33 150.0
1.19 120.0
1.03 90.0
0.77 50.0
v = v0 + at.
1. A small sailboat is observed to move directly away from a dock. Its distance from the dock at various times is measured by observing the time it passes each of several buoya along its route. From the graph below determine the velocity of the boat. How long after the boat left the dock was the timer started?
2. Two women walk out of the clubhouse climb into a golf cart and set off at constant speed on a straight path over smooth level ground. Along the way, the passenger starts a stopwatch and begins to record the time they pass certain markers along their route. Each marker gives the distance between the marker and the clubhouse. The data are given in the graph. From the graph determine the velocity of the cart. Write an equation giving the position s of the cart relative to the clubhouse in terms of the time t shown on the stopwatch. How long after they left the clubhouse did they start the stopwatch? How far are they from the clubhouse when the stopwatch reads exactly 7 minutes?
smi 6
.
4 .
.
2 .
0 5 10 15
t min.
3. A bicycle rider leaves the courthouse in her village and pedals at a constant speed along a straight level road to the courthouse in the next town which is thirty miles away. In the afternoon she begins to notice the time according to her wristwatch when she passes certain mileposts along the way. These posts give the distance from her courthouse. The data are shown graphically below. From the graph determine the speed in mi/min at which the rider travels. To the nearest minute according to her watch at what time did she leave the courthouse?
m 30
i
l 20
e
p 10
o
s 0 1:00 2:00 3:00
t wristwatch time
n
u
m
b
e
r
4. A car is traveling at a constant speed of 25 mi/h. Can you say that the car is not accelerating? Explain clearly.
5. Assume that the earth moves at constant speed in a circular orbit around the sun. Estimate the orbital speed of the earth.
Answer 108,000 km/h.
6. A horse-drawn sleigh is driven east over a straight level road which passes through two villages 21 miles apart. Easton is due east of Weston. The driver notes the time on a stopwatch that she passes certain mileposts which give the distance from the Post Office in Weston. The data are shown in the graph. From the graph determine where the sleigh was relative to the Weston Post Office, when the driver started her stopwatch. Write your answer in a complete sentence. From the graph obtain an equation that gives the position s of the sleigh at time t. Use your equation to find the time according to her stopwatch that she reaches Easton. Explain clearly and completely your reasoning in arriving at each answer.
7. To a good approximation Venus travels at constant speed in a circular orbit around the sun. Calculate the distance Venus moves in one day along its orbital path.
Answer 3 million km.
8. A body is experiencing a constant acceleration of 3 m/s2. How much speed does the body gain each second? If it starts from rest, what is its speed after 2 seconds? What is its speed after 5 seconds?
9. A bicycle rider increases her speed from 5 to 10 mi/h in the same time that a car goes from 50 to 55 mi/h. Which, if either has the greater acceleration? Explain clearly your reasoning.
10. An air track is inclined slightly with respect to the horizontal so that a glider starting from rest will slide with constant acceleration over the full length of the track 1.30 m in 3.21 seconds. Calculate the speed of the glider as it reaches the lower end of the track.
Answer 0.81 m/s.
11. Starting from rest a drag racer reaches a speed of 240 km/h over a distance of 420 m on a straight level track. Calculate the acceleration of the racer.
Answer 5.29 m/s2.
12. The takeoff speed of a certain airplane is 155 km/h. Starting from rest the aircraft requires a run of 750 m to become airborne. How much time is required for this takeoff roll?
Answer 34.8 s.
13. A go-cart starts from rest on a straight level strip and accelerates uniformly until it reaches its top speed of 58 km/h fourteen seconds later. How far from its starting point has the go-cart traveled one minute after it started moving?
Answer 854 m.
14. A rocket-propelled sled traveling with a speed of 327 km/h is brought to rest uniformly over a distance of 140 meters. Calculate the stopping time and the acceleration.
Answer 3.08 s; 29.5 m/s2.
15. on a sled slides down a long straight slope. The position relative to some point on the slope is observed at various times. The data are shown graphically below. From the graph determine the acceleration of the sled. Write an equation for the position s of the sled at time t. Assuming that the motion continues with the same acceleration use your equation to find the position of the sled when the timer reads 80 seconds.
16. A car traveling 60 km/h can be stopped in a distance of 29.3 meters. How much distance is required to stop the car if it is traveling 135 km/h typical speed on a German autobahn?
Answer 148 m.
17. A rocket is launched vertically upward and reaches a maximum speed of 4,800 km/h at an altitude of 240 km at which point its engine is cut off. Find the time it takes the rocket to reach this altitude and the average acceleration in m/s2 of the rocket.
Answer 6 min; 3.70 m/s2.
18. A passenger train sits on a straight level track with the locomotive sitting some distance from the terminal. The train begins to move with a constant acceleration along the track. At the instant that it begins to move the engineer starts a stopwatch. Position markers indicating the distance from the terminal are located at intervals along the track. After some time the engineer begins to note the time according to the stopwatch at which he passes certain of these markers. His data are shown in the graph below. How fast in km/h is the train traveling when the locomotive is 1,200 m from the terminal? How far from the terminal was the locomotive when the train started moving?
position meters 400 .
.
200 .
.
0 .2 .4 .6 .8
time 2 minutes 2.
19. A car is traveling on a straight level road at a speed of 72 km/h. The driver suddenly hits the brake and stops in 14.3 seconds. How far does the car travel in this time?
Answer 143 m.
20. An aircraft carrier is sailing with a speed of 27 km/h with no wind. Assisted by a horizontal catapult an aircraft is launched over the bow of the ship. The uniform acceleration of the aircraft is 34.3 m/s2. Its airspeed as it leaves the catapult is 168 km/h. Calculate the length of the catapult.
Answer 30.9 m.
21. A small block moving with a constant acceleration is observed at one instant to be moving to the right with a speed of 4.87 m/s. It is seen 4.17 s later 2.46 m to the right of the position where it was first observed and now moving to the left. How fast is the block moving when this second observation is made?
Answer 3.69 m/s.
22. A bullet is fired with a velocity of 400 m/s into a wooden door that is 2.68 cm thick. The bullet passes clear through the door and emerges from the other side 8.70 x 10 -5 second after it first struck the door. How fast is the bullet moving as it leaves the door? Assume that the bullet changes speed uniformly as it passes through the door.
path of bullet >
door
Answer 216 m/s.
23. A car on a straight stretch of road is traveling at 135 km/h. The driver suspects a radar trap ahead and slows his car uniformly to 88 km/h in seven seconds. How far does the car travel in this time?
Answer 217 m.
24. A windsurfer is sailing along at 12.8 km/h when his sail catches a sudden gust of wind and increases his speed uniformly to 17.3 km/h in 11.6 seconds. How far does the gust carry him in this time?
Answer 48.5 m
25. A particle moving with a constant acceleration is observed at one instant to be moving to the left with a speed of 13.8 m/s. Eight seconds later it is observed to be moving to the right with a speed of 22.7 m/s. How far apart are the two points where the particle was observed? Is the second point to the right or to the left of the first point?
Answer 35.6 m to the right.
26. The pilot of an aircraft in level flight on a straight course reduces the aircraft speed uniformly from 183 km/h to 152 km/h in seventeen seconds. How far does the aircraft travel during this time interval?
Answer 791 m.
27. A small block slides along a straight path on a rough horizontal surface is observed at one instant to be moving to the right with a speed of 0.932 m/s. Five seconds later it is still moving to the right but it has slowed down to 0.560 m/s. Assuming the acceleration is constant how far does the block travel in this time interval?
Answer 3.73 m.
28. On an interplanetary journey, a spacecraft is moving with a constant velocity of 17,100 km/h. The engine is fired for 46.9 s accelerating the craft uniformly to 17,850 km/h without changing the direction of motion. How far does the spacecraft travel during the time the engine is running?
Answer 227.7 km.
29. Starting from rest a small cart is pulled along a straight track. At various times t observed on a stopwatch the position s of the cart is measured relative to one end of the track. The data are given in the graph. From the graph obtain an equation for the position s in terms of the time t. Use this equation to find where the cart is located relative to the end of the track and how fast it is moving when the stopwatch reads 1.68 s.
30. A girl on a sled is given a push that starts her down a long straight slope with an initial speed of 0.833 m/s. She reaches the bottom of the slope 23.8 s later with a speed of 1.944 m/s. Assuming that the acceleration of the sled is constant what is the length of the slope?
Answer 33.0 m.
31. A car starts from rest with a constant acceleration of 1.87 m/s2. When it reaches a speed of 70 km/h it stops accelerating and continues traveling at this constant speed. How long does it take the car to travel one kilometer from its starting point?
Answer 56.6 s.
32. A deputy sheriff is traveling east along a straight stretch of narrow two-lane highway at a constant speed of 85 km/h. In her rearview mirror she sees a small car coming up behind her at high speed. As the car whizzes past her it is traveling at its top speed of 145 km/h. The county line intersects this highway 5.5 km east of the point where the speeder draws even with the deputy. The deputy has no authority to issue a speeding ticket unless she apprehends the culprit within her own county. If he makes it over the county line before she can stop him, she will not be able to give him a speeding ticket. As the speeder draws even with her, the deputy begins to accelerate uniformly at 0.445 m/s2 until she overtakes him. She shoots past the speeder, whips in front of him, and drives down the middle of the road so that he cannot pass. They both immediately begin to reduce speed at uniform rates until they stop. The distance from the point where she overtakes him to the point where they stop is 1,350 meters. Does the speeder get a ticket? How fast was the deputy traveling when she passed the speeder?
33. A sailboat leaves its dock and sails in succession to several points on the shore of a large lake. To reach the first point the boat sails 1.43 mi in a direction 37 N of E. From there it heads 26 S of W to a point 2.37 mi away. The last leg of the journey is over a distance of 1.64 mi at 70 N of E. By means of a scale drawing find the displacement of the boat relative to its dock.
Answer 1.43 mi 72 N of W.
34. A military tank starts at its home base and travels 14.4 mi in a direction 62 N of E. The tank then heads 17 S of W and travels 11.2 mi in that direction. Finally it turns and travels 21.9 mi in the direction 36 S of E. From a scale drawing determine the resultant displacement of the tank from its home base.
Answer 14.2 mi 14 S of E
35. A hiker sets out from his base camp early in the morning and travels 8.5 mi in the direction 17 N of E and stops for lunch. After lunch he heads 75 N of W and travels 7.4 mi in that direction before he stops to rest. After resting for a while he picks up his gear and walks 6.5 mi in the direction 14 N of E. Find the hiker's resultant displacement magnitude and direction from his base camp.
Answer 16.8 mi 41.8 N of E.
36. A lone horseman leaves the village of Polena and travels to Ascoti which is 4.96 mi away in the direction 21.3 N of E. At Ascoti he turns and heads in the direction 40.6 N of E toward Tonio 3.10 mi from Ascoti. He leaves Tonio traveling in the direction 28.7 S of W and rides for 7.95 miles. At this point darkness overtakes him and he stops for the night. All of the roads between the villages are straight paths over level ground. How far does he travel on this journey? At the end of the journey where is the rider relative to Polena? Write your answer in a complete sentence.
37. Sorgenloch is 2.3 km due south of Nieder Olm. Stadecken is 3.2 km due west of Nieder Olm. Essenheim is 1.9 km in a direction 22 E of N from Stadecken. Find the location of Essenheim relative to Sorgenloch.
Answer 4.8 km 58 N of W.
Chapter 3
1. If light is incident on a mirror from the left on which side left or right of the mirror is a real image formed? From a physical point of view why ia this so?
2. A object 1.8 mm tall is placed 58.0 cm to the left of a small concave mirrror with a focal length of 26.0 cm. Find the position height, and character of the image.
Answer 47.1 cm to the left of the mirror 1.5 mm real, inverted.
3. A object 1.1 mm tall stands 31.3 cm in front of a concave mirror with a focal length of 56.2 cm. By means of a scale drawing find the position height, and character of the image.
Answer 70.6 cm behind the mirror, 2.5 mm, virtual, erect.
4. The focal point of a small convex mirror is 36.2 cm from the mirror. What is the location of the image of a small object placed 41.7 cm in front of this mirror?
Answer 19.4 cm behind the mirror.
5. Find the position, height, and character of the image of an object standing in front of a small convex mirror. The focal point is 19.3 cm from the mirror. The object is 1.3 mm tall and is 40.7 cm from the mirror.
Answer 13.1 cm behind the mirror, 0.4 mm, virtual, upright.
6. A small object stands 49.1 cm to the left of a small spherical mirror. The image formed by the mirror is 29.1 cm to the right of the mirror. By means of a calculation determine what kind of mirror this is.
7. A object 2.9 mm tall stands 34.2 cm to the right of a convex mirror with the focal point 23.7 cm from the mirror. By means of a scale drawing find the position, height, and character of the image.
Answer 14.0 cm to the right of the mirror, 1.2 mm, virtual, erect.
8. A small spherical mirror produces a virtual image of a very small object that stands 23.7 cm in front of the mirror. The image is 2.7 times as tall as the object. In one sentence give as complete a description as possible of this mirror.
9. A small spherical mirror produces a real image of a very small object located 21.9 cm in front of the mirror. The image is 3.5 times as tall as the object. As completely as you can describe this mirror.
10. A small spherical mirror produces a virtual image of a very small object located 21.9 cm in front of the mirror. The image is 3.5 times as tall as the object. As completely as you can describe this mirror.
11. A small object stands 26.4 cm in front of a certain mirror. An image of the object is formed 38.9 cm behind the mirror. By means of a calculation determine whether this is a concave mirror or a convex mirror.
12. A object placed 39 cm in front of a small spherical mirror forms a image 51 cm behind the mirror. Give a quantitative description of the mirror.
13. Explain clearly what is meant by refraction. What is the physical reason refraction occurs?
14. Why does a lens have two focal points whereas a mirror has only one? Give a clear and complete physical explanation.
15. Explain clearly and completely why we say that a real image is found in front of a mirror but a real image is formed behind a lens.
16. A small object 2.80 mm tall stands 43.0 cm to the left of a thin diverging lens with a focal length of 35.0 cm. Use the lens formula to find the position, height, and character of the image formed by this lens. Write your answers in a complete sentence.
Answer 19.3 cm to the left of the lens, 1.26 mm, virtual, upright.
17. A thin converging lens has a focal length of 27 cm. A small object is placed 35 cm to the right of this lens. Use the lens formula to determine the location of the image relative to the lens.
Answer 118 cm to the left of the lens.
18. The focal points of a thin diverging lens are located on either side of the lens on its principal axis at a distance of 51 cm from the lens. Use the lens formulas to find the position, height, and character of the image the lens forms of an object 4.20 mm tall standing 22 cm to the left of the lens.
Answer 15.4 cm to the left of the lens, 2.93 mm, virtual, upright.
19. A small object 2.8 mm tall stands 39.0 cm to the left of a thin diverging lens with a focal length of 75.0 cm. Use the lens formulas to find the position, height, and character of the image formed by this lens. Write your answers in a complete sentence.
Answer 25.6 cm to the left of the lens, 1.8 mm, virtual, erect.
20. Find the position, height, and character of the image of a small object 4.8 mm tall that stands 41.0 cm to the right of a thin converging lens with a focal length of 52.8 cm. Solve this problem first by using the lens formulas then choose appropriate scales vertical and horizontal and solve it by means of a scale drawing.
Answer 183 cm to right of lens, 2.15 cm, virtual, erect.
21. A object 1.12 mm high is located 30.0 cm in front of a converging lens which has a focal length of 40.0 cm. Calculate the position and height of the image. Is the image real or virtual? Is it upright or inverted?
Answer 120 cm in front of the lens, 4.48 mm, virtual, upright.
22. The focal points of a diverging lens are located on the principal axis 37.0 cm on either side of the lens. A object 0.75 cm tall is placed 60.0 cm to the left of this lens. By means of a scale drawing find the position and height of the image in the laboratory. Is the image real or virtual? Is it erect or inverted?
Answer 22.9 cm to the left of the lens, 0.29 cm, virtual, erect.
23. A small object 2.1 mm tall stands 33.0 cm to the left of a thin converging lens with a focal length of 68.0 cm. By means of a carefully drawn scale drawing find the position, height, and character of the image formed by this lens.
Answer 64 cm to the left of the lens, 4.1 mm, virtual, erect.
24. Very large telescopes are expensive to build and operate. Explain clearly and completely why it is desirable to build large astronomical telescopes.
25. Why is a auxiliary mirror needed in small reflecting telescopes?
26. What are the disadvantages of building large refracting astronomical telescopes compared to building large reflectors?
27. Two converging lenses are separated by a distance of 63.0 cm measured along the principal axis of these lenses. Each has a focal length of 21.0 cm. A small object 0.370 cm tall stands 57.0 cm to the left of this system. Find the position, height, and character of the image formed by this system. Solve this problem first by means of a scale drawing then by means of the thin lens formulas.
Answer 71.4 cm to right of second lens, 0.518 cm, real, erect.
28. A converging lens with a focal length of 42 cm stands 15 cm to the left of a second converging lens which has a focal length of 27 cm. A object 0.28 cm high stands 17.8 cm to the left of this system. How tall is the image produced by this system? Where is the image located? Describe the character of the image. Solve this problem by using the formulas and then by using a scale drawing.
Answer 0.694 cm, 65.6 cm to right of second lens, real, inverted.
29. A object 0.44 cm tall is placed 72.0 cm to the left of a thin converging lens which has a focal length of 17.2 cm. A diverging lens with focal points 22.7 cm from the lens stands 33.3 cm to the right of the converging lens. First use a scale drawing then use the lens formulas to determine the location, the height, and the character of the final image produced by this optical system.
Answer 7.27 cm to the left of the second lens, 0.939 mm, virtual, inverted.
30. A thin converging lens is placed 23.6 cm to the left of a thin diverging lens. The magnitudes of the focal lengths of these lenses are 52.3 cm and 19.9 cm respectively. A small object stands 5.2 mm tall at a distance of 176.0 cm to the left of the converging lens. Use the thin lens formulas to determine the location, height, and character of the image of this object. Do not attempt to make a scale drawing for this problem.
Answer 32.7 cm to the left of the diverging lens, 1.4 mm, virtual, erect.
31. A telescope consists of two converging lenses a objective with a focal length of 0.83 m and a eyepiece with a focal length of 27 mm. The instrument is used to observe a small object located 17.0 m in front of the objective. Find the distance between these lenses if the final image is real and is formed at infinity. Solve this problem by means of the formulas only. Do not make a scale drawing.
Answer 89.96 cm.
32. A small concave mirror and a thin diverging lens are placed 25.7 cm apart as measured along their principal axes. The magnitudes of the focal lengths of these elements are 16.2 cm and 36.5 cm respectively. A small object 2.3 mm tall is placed between the two elements at a distance of 24.3 cm from the mirror. Light from the object reflects off the mirror and passes through the lens to form an image. Use the formulas to find the position, height, and character of the image formed by this optical system.
Answer 61.4 cm behind the lens, 12.3 mm, real, inverted.
33. A thin converging lens with a focal length of 21.5 cm stands 25.0 cm to the left of a thin diverging lens of focal length - 36.4 cm. A small object 0.37 mm tall is placed 38.7 cm to the left of the converging lens. Use the lens formulas to find the position, height, and character of the image formed by this system.
Answer 65.3 cm to the right of the diverging lens, 1.29 mm, real, inverted.
34. A thin diverging lens stands to the left of a thin converging lens. A small object 2.2 mm tall is placed to the left of the diverging lens as illustrated in the scale drawing below. By carefully completing the scale drawing find the position, height, and character of the image in the laboratory formed by this lens system. Give your answer in a complete sentence.
diverging lens converging lens
object
F1 F2 F2 F1
<50 cm>
Answer 46 cm to the right of the converging lens, 3 mm, real, inverted.
35. A small object is placed at a point A on the principal axis of a small thin lens. The image formed by the lens is four times as tall as the object. The object is moved 10 cm farther away from the lens to a point B on the principal axis and again the image is four times as tall as the object. How far from the lens is the object when it is at point B?
Answer 25 cm.
36. By means of words and a drawing explain clearly why chromatic aberration is objectionable in a refracting telescope.
37. Light falls on a glass prism as shown in the drawing. Based on the drawing which color light if either travels faster in the glass, yellow or green? Explain clearly and completely how you can tell.
Chapter 4
1. A force of 12 N is applied to each of two bodies A and B. The resulting acceleration of A is 15 m/s2 while B accelerates at 19 m/s2. Which body if either has the greater inertia?
2. A little boy sitting in a toy wagon tumbles backward out of the wagon when the wagon is suddenly jerked forward. In terms of inertia explain clearly why the boy falls backward.
Because the little boy may had not been holding onto the sides or something.
3. A horizontal force of thirteen newtons is exerted on a thirty kilogram crate in pushing it across a smooth horizontal floor. How far does the crate move in four seconds if it starts from rest?
Answer 3.47 m.
4. Lead is more dense than aluminum. Each of two balls one aluminum the
other lead has a mass of 1.32 kg. Which if either has the greater inertia? Explain.
aluminum 2,650 density kg/m3
lead 11,370
5. A 1.847 kg metal cylinder is 13.73 cm long and has a diameter of 4.38 cm. Calculate the density of the cylinder and identify the metal from the list in the table.
density
metal kg/m3
aluminum 2,650
brass 8,600
copper 8,930
gold 19,320
lead 11,370
steel 7,380
zinc 7,150
6. A rectangular block of gold measures 16.5 mm x 11.3 mm x 31.9 mm and a rectangular block of lead 14.5 mm x 13.6 mm x 64.2 mm. Which of these if either has the greater inertia. See table in previous exercise. Explain clearly your reasoning.
7. Assume that the earth, the sun, and the moon are spheres and calculate the mean density of each.
Answer 5,520 kg/m3, 1,410 kg/m3, 3,340 kg/m3.
8. It is expected that late in its life the sun will evolve to a very dense compact object called a white dwarf. Typically a white dwarf star has a mass of the order of that of the sun packed into a volume the size of the earth. Use the data in the appendix to estimate the density of a typical white dwarf. From your result estimate the mass of a teaspoonful of this matter. Approximately how much would a teaspoonful of this matter weigh on earth?
Answer 2 x 10 9 kg/m3, 7,000 kg, 8 tons.
9. A three engine 196,000 kg DC-10 requires 3,320 meters of runway to reach its takeoff speed of 310 km/h. Find the average net thrust provided by each engine during the takeoff roll.
Answer 73,000 N.
10. Starting from rest a man pushes a 380 lb crate a distance of twelve feet in 4.7 seconds over a rough horizontal floor. Assuming constant acceleration and a frictional force of 95.0 lb find the horizontal force that the man exerts on the crate.
Answer 107.9 lb.
11. A constant total force of 1.79 N is applied to a certain object. The position s of the object is observed at various times t. The results are displayed in the graph. From the data given calculate the mass of the object.
s m 60 .
40 .
20 .
00 100 200 300
t2 s2
12. The brakes of a 1,200 kg truck supply a average force of 4,200 N in
stopping the truck. How far does the truck travel before stopping if its speed is 90 km/h when the brakes are applied?
Answer 89.3 m.
13. A small cart is pulled by a constant force along a flat horizontal track. After the cart is set in motion its position relative to the left end of the track is observed at various times according to a electronic timer. From the graph determine the acceleration of the cart. Obtain a equation for the position s of the cart at time t. Assuming that the motion continues with this same acceleration use your equation to find the position of the cart relative to the left end of the track when the timer reads 0.875 second.
14. What force is required to give a 0.49 kg football an acceleration of 400 m/s2? If this force is applied over a distance of 1.70 m what is the speed with which the ball leaves the passer's hand?
Answer 196 N, 36.9 m/s.
15. A catapult for launching aircraft from a aircraft carrier exerts a constant horizontal force of 328,000 N on a 11,300 kg aircraft. Through what distance must this force act to give the plane a speed of 220 km/h?
Answer 64.3 m.
16. A girl is lying on a sled which is gliding with a constant velocity of 4.53 m/s over perfectly smooth ice. The combined mass of the girl and the sled is 38.7 kg. The sled enters a patch of rough ice over which the speed is slowed to 2.19 m/s. On the rough ice the girl and sled experience a constant frictional force of 12.8 N. How far did the sled travel on the rough ice?
Answer 23.8 m.
17. A small 437 g block is set in motion with a speed of 7.12 m/s on a rough horizontal surface over which it experiences a constant frictional force of 1.68 N. How fast is the block sliding 0.78 s after it is released? How far does it slide in this time interval?
Answer 4.12 m/s, 4.38 m.
18. A artillery shell explodes in flight. Neglect air friction and discuss the motions of the pieces of the shell before and after the collision.
19. Two gliders are free to move on a horizontal air track as in Fig. 4.5. The masses of these gliders are m = 0.311 kg and m2 = 0.439 kg. The 0.311-kg glider moving to the right with a speed of 0.532 m/s collides with the second glider which is at rest. The two gliders stick together. What is the velocity of the 0.439 kg glider after the collision?
Answer 0.221 m/s to the right.
20. Suppose the two gliders in the previous exercise do not stick together and after the collision the 0.439 kg glider moves to the right with a speed of 0.427 m/s. Calculate the velocity of the 0.311 kg glider after the collision.
Answer 0.071 m/s to the left.
21. A small spacecraft of mass 267 kg on a interplanetary flight is traveling with a speed of 67,317 km/h. Its engine is fired briefly for 2.3 seconds. During this time interval the speed of the craft is increased uniformly by ejecting the hot gas of the burned fuel. During the burn 1.70 kg of hot gas is ejected with a speed of 2,150 m/s relative to the spacecraft. Calculate the speed of the spacecraft at the end of the burn. Calculate the thrust supplied by the engine during the burn. Assume that the mass of the spacecraft remains constant.
Answer 67,366 km/h, 1,589 N.
22. A 79 kg lifeguard dives from a 160 kg rowboat initially at rest. The lifeguard leaves the boat with a horizontal speed of 3.44 m/s. Calculate the recoil speed of the boat. Neglect the transfer of momentum to the water.
23. A railroad car weighing 38 tons is loaded with nine tons of logs. It rolls along a straight level track at 8.37 ft/s until it collides with a identical car at rest on the same track with a 23 ton load of logs. The two cars couple together in the collision. What is the velocity of the first car after the collision? Explain clearly your reasoning in arriving at your answer.
24. If you drop a object from rest and allow it to fall freely what is its acceleration at the end of two seconds?
25. Suppose you throw a ball vertically upward. The ball will rise to some maximum height then fall back to the ground. What is the velocity of the ball when it reaches maximum height? Explain. What is the acceleration of the ball at maximum height? Explain.
26. If you drop a steel ball its acceleration toward the ground is about 32 ft/s2. Suppose you threw the ball toward the ground. Would its acceleration after you released it be less than, greater than, or about equal to 32 ft/s2? Explain?
27. A book rests on a horizontal table. Make a free body diagram showing all of the forces acting on the book. Beside each vector write the name of the physical object exerting the force. Give the reaction force in the sense of Newton's third law to each of these forces.
28. The acceleration due to gravity on the moon is about one sixth of the value on the earth. On the earth the mass of a 114 lb woman is about 52 kg. Estimate her mass if she were on the moon. Estimate her weight if she were on the moon. Explain clearly and completely your reasoning in each case.
29. On the earth the mass of a 176 lb man is 80 keg. If the man were floating freely in space far from any large gravitating bodies planets, stars, etc. what would be his mass out there? What would be his weight out there? Explain clearly your reasoning.
30. A woman stands in a elevator that is moving with a downward acceleration of 4 ft/s2. Draw a free body diagram showing all of the forces exerted on the woman. Which if either of these forces is greater? Explain. In the sense of Newton's third law of motion what is the reaction force to each of these forces? From the information given can you tell in which direction the elevator is moving? Explain.
31. A 1,320 kg rocket is launched vertically upward. The engine supplies a constant thrust of 18,000 N until it is cut off when the rocket reaches a speed of 2,300 km/h. What is the altitude of the rocket when the engine is cut off? Neglect the effects of air friction and assume that the mass of the rocket remains constant.
Answer 53.2 km.
32. Starting from rest and accelerating uniformly a 68 kg man slides down the full length of a four meter rope in 1.10 seconds. How much force does he exert on the rope while he is sliding down?
Answer 217 N.
33. A string is tied to a four kilogram metal slug that rests on a workbench. The tension in the string must not be greater than 44 N or the string will break. By jerking upward on the string what is the largest acceleration that can be given to the slug without breaking the string?
Answer 1.2 m/s2.
34. A bell boy carries a 35 lb piece of luggage into a hotel elevator. How much force must he exert on the luggage handle to support it when the car moves with a upward acceleration of 9.8 ft/s2?
Answer 45.7 lb.
35. By means of a light string a ball hangs suspended from the roof inside a van moving on a straight level road. The string is observed to be inclined at a constant angle with respect to the vertical as shown in the drawing. Draw a carefully labeled free body diagram showing all of the forces exerted on the ball. State clearly the reaction force in the sense of Newton's third law of motion to each of these forces. Describe clearly the motion of the van. Can you tell in which direction the van is moving? If so give the direction and explain clearly your reasonong. If not explain why not.
36. A 6.43 kg rocket launched vertically upward from a stationary pad accelerates uniformly to a speed of 187 km/h which it reaches at a altitude of 621 m. Calculate the thrust of the engine during the rise. Neglect frictional effects.
Answer 77.0 N.
37. A 146 lb weight watcher stands on a spring scale in a elevator that is accelerating vertically downward at 2.38 ft/s2. What is the reading on the scale? Explain clearly.
Answer 135 lb.
38. A 117 lb woman stands in a hotel elevator which is slowing down at a rate of 1.98 ft/s2. Draw a clearly labeled free body diagram showing all of the forces exerted on the woman. Identify clearly the material body exerting each of these forces. Can you tell which force is greater? If so which? If not why not? In the sense of Newton's third law of motion identify clearly the reaction force to each of these forces.
39. A ball is seen to be moving vertically upward with a speed of 15.7 m/s. Some time later it is 22.6 m above the ground and moving straight down toward the earth's surface with a speed of 11.4 m/s. How high above the ground was the ball when it was first observed? Neglect the effects of air friction.
Answer 16.6 m.
40. A small two stage rocket is launched vertically upward from its pad. When the rocket reaches a altitude of 3.27 km where its speed is 1,070 km/h the burned out first stage is released and falls back to the earth. At this altitude the second stage engine is fired giving the rocket a constant upward thrust of 681 N. The total mass of this second stage is 25.7 kg. When the second stage speed reaches 2,250 km/h the engine cuts off. Find the altitude at which the engine of second stage cuts off. Neglect effects due to friction.
Answer 12.3 km.
41. Two identical 2 kg balls are connected by a light string. With another light string the system is suspended from the ceiling of a elevator that is accelerating upward at 1.3 m/s2. The arrangement is illustrated in the drawing. Calculate the tension in each string.
Chapter 5
Exercises
1. A little boy pulls with a constant horizontal force of 58 N on the handle of a wagon which his friend sits. He gives his friend a 38-m ride along a horizontal sidewalk. The combined mass of the rider and the wagon is 39.5 kg. How much work is done on the wagon by the boy pulling on it?
2. By applying a horizontal force of 289 N a dock worker pushes a 1,220 N crate at a constant velocity of 1.17 m/s a distance of 10.3 m over a rough horizontal floor in 8.80 seconds. Calculate the work done on the crate by the dock worker.
Answer 2,980 J.
3. A frictionless block and tackle is used to raise a 318 kg steel beam at constant speed from the ground to a height of 34.6 m. How much work is done in lifting the beam?
Answer 108,000 J.
J is joules.
4. A 384 kg rocket is launched vertically upward. The rocket accelerates uniformly until it reaches a speed of 1,320 km/h at a altitude of 37 km at which point the engine cuts off. While the engine is running it supplies a constant thrust of 4,460 N. Calculate the work done by the engine.
Answer 1.65 x 10 8 J.
5. Show clearly that Eq. 5.2 follows from Eq. 5.1 as outlined in the text.
6. A constant vertical force of 9,500 N is used to raise a 950 kg granite block from the ground to a height of twelve meters above the ground. Calculate the work done in lifting the block and the change in gravitational potential energy of the block. Are your two answers equal. Should they be? If so why? If not why not?
7. A 125 g ball is thrown vertically upward and rises to a maximum height of 6.37 m above the ground. During its rise the ball experiences a average frictional force of 0.183 N. Use the energy conservation principle to calculate the speed of the ball as it left the thrower's hand 2.13 m above the ground.
Answer 9.77 m/s.
8. A 41.8 kg boy sits on a 3.6 kg sled which is coasting with a constant velocity of 5.27 m/s over perfectly smooth level ice. The sled enters a patch of rough ice 17.8 m long over which it experiences a constant frictional force of 15.2 N. Use the energy conservation principle to find the speed at which the sled emerges from the rough stretch. Neglect the effects of air friction. What percent of the original energy of the boy and sled is lost due to friction on the rough patch of ice?
Answer 3.98 m/s, 42.9%.
9. A 39.34 kg girl starts from rest on a 3.29 kg sled and slides down a hill that is 8.62 m high. She experiences a constant frictional force over the full length of the 143.5 m slope. As she reaches the bottom of the slope her speed is 4.17 m/s. Calculate the combined frictional force she and the sled experience during the slide.
Answer 22.5 N.
10. A small 1.37 kg rocket is launched vertically upward from rest on the ground. The engine supplies a constant thrust of 24.25 N over a distance of 402 m at which point it runs out of fuel. As the rocket rises it experiences a average frictional force of 4.54 N. Use the energy conservation principle to calculate the maximum altitude reached by this rocket.
Answer 543 m.
11. A 2.46 kg block is set in motion with a initial velocity of 9.02 m/s on a rough horizontal surface. The combined effects of air resistance and the rough surface result in a constant frictional force of 6.82 N exerted on the block as it slides. Use the energy conservation principle to find how fast the block is moving when it is 3.19 m from the point at which it is released.
Answer 7.98 m/s.
12. Two balls one of mass m the other of mass M are both at rest. The same constant force F is applied to each ball separately over the same time interval At. At the end of the time interval which ball if either has the greater momentum? Which if either has the greater kinetic energy?
13. A 287 g ball attached to a string is set in motion in a vertical circle of radius 1.09m. The speed of the ball at the top of the circle is 5.66 m/s. Calculate the tension in the string when the ball is at the top of the circle. Use the energy conservation principle to find the speed of the ball at the bottom of the circle. Neglect frictional effects.
Answer 5.62 N, 8.65 m/s.
14. A 0.189 kg ball is thrown vertically downward from the top of a building that is 22.9 m tall. The speed of the ball as it leaves the thrower's hand is 8.75 m/s. Air resistance results in a average frictional force of 0.652 N on the ball as it travels downward. Use the energy conservation principle to find the speed of the ball at the instant before it hits the ground.
Answer 19.2 m/s.
15. A windsurfer is sailing along at 3.56 m/s when a sudden gust of wind hits his sail with a constant force of 392.0 N and continues over a distance of 42.7 m. During this run he and his board experience a constant frictional force of 371.9 N. The total mass of the surfer and his board is 82.4 kg. Use the energy conservation principle to calculate the speed of the surfer at the end of the 42.7 m run.
Answer 5.79 m/s.
16. A gun is aimed vertically upward and fired. The 4.22 g bullet leaves the barrel with a speed of 163 m/s. As the bullet rises it experiences a average frictional force of 0.0179 N. How high above the end of the barrel of the gun does the bullet rise?
Answer 946 m.
17. A 38.15 kg girl sits on a 2.32 kg sled at rest at the top of a straight slope that is 33.5 m long and 9.1 m high. The girl's sled is pushed downhill with a constant force of 120 N parallel to the slope. The pushing of the sled has stopped after the sled has moved 3.22 m along the slope. The girl and her sled continue coasting to the bottom of the slope. The frictional force they experience during the slide is 83.6 N. Use the energy conservation principle to find the speed of the sled when it reaches the bottom of the slope.
Answer 7.68 m/s.
18. A small boy sits in a soap box racer at rest at the top of a straight slope that is 87.3 m long and 18.2 m high. The mass of the boy and the racer combined is 42.3 kg. With a constant force applied parallel to the slope. The boy is pushed the racer downhill. The pushing has stop after the racer has moved 2.75 m along the slope. The racer continues coasting downhill until it reaches the bottom of the slope where it has a speed of 4.80 m/s. The constant frictional force the boy and the racer experience during the downhill ride is 84.2 N. Use the conservation of energy principle to calculate the force of the exerts on the racer while the racer is being pushed.
Answer 107 N.
19. By blowing with the mouth on the lower end of a straight hollow tube a small wooden ball is projected from the upper end of the tube as illustrated. The mass of the ball is 28.4 g. The ball leaves the tube with a speed of 5.38 m/s. Calculate the total energy gained by the ball as it travels along the length of the tube.
20. A rough track consists of a straight horizontal section and a curved section oriented in a vertical plane. A small 119 g block is released from rest at the top of the curved section as illustrated. The block reaches the horizontal section with a speed of 3.65 m/s. The upper end of the curved part of the track is 87.3 cm above the horizontal part. Along the horizontal part of the track the block experiences a constant frictional force of 1.185 N. Use the conservation of energy principle to find the energy lost by the block due to friction on the curved part of the track and how far the block slides on the horizontal part.
Answer 0.225 J, 66.9 cm.
21. A 227 g glider moves along a straight horizontal air track with a speed of 0.417 m/s. It strikes a second glider at rest. The two gliders stick together and move with a velocity of 0.238 m/s in the original direction of the first glider. Find the mass of the second glider. How much energy did the gliders lose to the environment in the collision?
Answer 171 g, 8.47 x 10 -3J.
22. A particle of mass 1.67 x 10 -27 kg moving to the right with speed 9.380 x 10 6 m/s collides head on with a second particle of mass 6.680 x 10 -27kg moving to the right with a speed of 0.260 x 10 6 m/s. After the collision the second particle is still moving to the right with speed 2.345 x 10 6 m/s. Find the speed and direction of motion of the first particle after the collision. By means of a calculation determine whether this is a elastic collision.
Answer 1.040 x 10 6 m/s to the right, inelastic.
23. A 0.783-kg mass moving to the right with a speed of 5.21 m/s collides head on with a 1.223 kg mass moving to the left with a speed of 4.07 m/s. After the collision the 1.223 kg mass is moving to the right with a speed of 1.130 m/s. Find the velocity magnitude and direction of the 0.783 kg mass after the collision. What fraction of the original energy of the system is lost in the collision? Explain clearly what happened to this energy.
Answer 2.91 m/s to the left, 80.2%.
24. A 813 g ball moving to the right with a speed of 5.08 m/s collides head on with a 1.228 g ball moving in the opposite direction with a speed of 3.88 m/s. After the collision the 813 g ball is moving to the left with a speed of 4.17 m/s. From a calculation determine whether the collision is elastic or inelastic.
Answer inelastic.
25. A 142 g hockey puck sliding with a speed of 36.4 m/s to the right on smooth ice collides head on with a second puck at rest. After the collision the 142 g puck is moving to the left with a speed of 1.74 m/s and the other puck is moving to the right with a speed of 29.7 m/s. Find the mass of the second puck. What percent of the total energy was lost by the two puck system in the collision?
Answer 182 g, 14.3%.
26. Two blocks equipped with bumpers are free to slide on a smooth horizontal surface. The block of mass 1.837 kg slides with a velocity of 9.25 m/s toward the right. It collides head on with the second block which has mass 0.722 kg sliding toward the left with a speed of 3.84 m/s. After the collision the second block is seen to be moving to the right with a speed of 6.54 m/s. Calculate the velocity of the first block after the collision. What percent of the initial kinetic energy of the system is lost in the collision?
Answer 5.17 m/s to the right, 52.3%.
27. Estimate the earth's kinetic energy due to its orbital motion around the sun.
28. The Electric Company charges customers for energy at a rate of 8.2 cents per kilowatt hour. A automatic timer is set to switch on a 250-W outdoor flood lamp at 6:00p.m. each day. At 11:30 p.m. each night the timer automatically switches off the lamp. Estimate the monthly cost of operating this lamp.
Answer $3.38.
29. A small aircraft is cruising in level flight with a constant speed of 180 km/h. The aircraft is experiencing a constant drag of 930 N due to air friction. Calculate the power the engine must deliver to overcome the drag.
Answer 46.5 KW.
30. It takes exactly one minute for a winch to drag a heavy crate at constant speed a distance of 11.7 m over a rough horizontal floor. The tension in the winch cable as it pulls horizontally on the crate is 2.930 N. Calculate the power delivered to the crate by the winch.
31. A 1,290 kg automobile starts from rest and accelerates uniformly until it reaches a speed of 110 km/h in 31.3 seconds. Calculate the power delivered to the wheels of the automobile in this run. Neglect frictional effects.
Answer 19.2 Kw.
32. A woman draws water from a deep well as shown in the illustration. The surface of the water is 27.4 m below the top of the well where the woman is standing. She hauls the water up by turning a windlass by means of a crank. The well rope passes over a pulley and winds around the turning windlass. It takes her 21.3 s to draw the filled bucket out of the well. She turns the crank so that the bucket rises at constant speed. The total mass of the water and the bucket is 25.5 kg. Calculate the power the woman delivers to the bucket of water in raising it. Suppose she uses a electric motor to turn the crank instead of turning it by hand. Assume that 70% of the energy the motor consumes actually goes to raising the bucket of water. At 8.2 cents per kWh how much would it cost to run the motor in bringing up each bucketful of water?
Answer 321 W, 0.022 cents.
Chapter 6
Exercises
1. The diameter of a old 78 rpm phonograph record is ten inches 24.5 cm. Calculate the centripetal acceleration of a point on the rim of this record when it is playing on the phonograph.
Answer 8.17 m/s2.
2. Assume the planet Mars moves with a constant speed in a circular orbit around the sun. Calculate the orbital acceleration in m/s2 of Mars.
3. On October 4, 1957 the USSR placed Sputnik the first artificial satellite in orbit around the earth. At a altitude of 575 km Sputnik followed a nearly circular orbit around the earth completing a orbit in 96 minutes. From the data calculate the centripetal acceleration of Sputnik.
Answer 8.26 m/s2.
4. A motorcycle on a circular track has a constant speed of 150 km/h. It takes 38.4 s for the motorcycle to make a complete circuit of the track. What is the centripetal acceleration of the motorcycle?
Answer 6.82 m/s2.
5. A cyclist is traveling at 24.3 km/h. The wheel of his bicycle has a diameter of 26 inches 66.0 cm. Calculate the centripetal acceleration of a small section of the tread of his front tire. How fast is the wheel turning at this speed?
Answer 138 m/s2, 195 rpm.
6. A pilot pulls his aircraft out of a dive along the arc of a large vertical circle. At the lowest point bottom of the dive what forces are exerted on the pilot? Name the material body that exerts each force. How are these forces related to the centripetal force on the pilot?
7. Referring to the previous exercise find the force the aircraft exerts on a 80-kg pilot if the speed of the aircraft at the bottom of the circle is 690 km/h and the radius of the circle is 2.030 meters.
Answer 2.232 N
8. A small ball is attached to a light string and set in motion in such a way it swings completely around in a vertical circle shown. At the bottom of the circle the tension in the string is 9.35 N and when the ball is at the top of the circle the tension is 2.06 N. Find the mass of the ball.
Answer 0.124 kg.
9. The earth describes a nearly circular orbit around the sun. The earth's orbital speed is about 1.075 x 10 5 km/h. Other relevant data are given in Appendix B. From these data estimate the average centripetal force exerted on the earth as it orbits the sun.
Answer 3.5 x 10 22 N.
10. A woman standing on the surface of the rotating earth at the equator experiences a centripetal acceleration. Making use of a clearly labeled free body diagram explain clearly and completely why the woman experiences a centripetal acceleration. Calculate the centripetal acceleration of the woman.
Answer 0.034 m/s2.
11. A 352-g ball hangs vertically at one end of a light string. The other end of the string is fixed at a point 73.4 cm directly above the ball. The ball is suddenly given a sharp horizontal blow which sets it in motion with a initial speed of 5.26 m/s. This will cause the ball to swing along a circular arc as illustrated. What is the tension in the string at the instant after the ball is struck? When the ball reaches a position at the same height as the point of suspension the string is stretched out horizontally. What is the tension in the string at this point? Will the ball swing all the way around or only part of the way?
Answer 16.7 N, 6.37 N.
12. Assume the moon describes a circular orbit around the earth. Calculate the corresponding centripetal acceleration of the moon.
13. In terms of centripetal force and centripetal acceleration explain clearly and completely why it is dangerous to drive a car too fast around a highway curve.
14. A 235-g steel ball is attached to one end of a light string. The other end of the string is fastened to the ceiling so the ball hangs 1.332 m directly below the point of suspension. The ball is pulled aside as indicated by the dashed line in the illustration to a point 1.252 m below the ceiling. It is released from rest at that point to swing back and forth as a pendulum. Calculate the tension in the string when the ball is at the lowest point in its swing. Neglect frictional effects.
Answer 2.58 N.
15. Use the orbital data for the planets given in Table 1.1 to calculate the centripetal acceleration a in AU/yr2 of each of the planets as it orbits the sun. Tabulate your results and note that as the period T increases the centripetal acceleration a decreases. Assume the relationship between a and T is a power law. That is
a = KTx3.
where the constant parameters x and K have the same values for all planets. If we take the logarithm of both sides of this equation we get
log a = x log T + log K.
Use your results for the planets to make a plot of log a versus log T. You should have a linear graph from which you can obtain values for x and K. Express x as the ratio of two small integers and rewrite the power law for a above using your values of x and K.
16. A hydrogen atom consists of a electron and a proton. According to a simple model of the atom the electron travels at constant speed in a circular orbit around the much heavier proton. The separation between the electron and proton is 5.28 x 10 -11m. The electron makes 6.60 x 10 15 complete orbits each second. Use these data and the information in Appendix B to calculate the centripetal force experienced by the electron in this model of the hydrogen atom.
Answer 8.27 x 10 -8 N.
17. A 1,230-kg automobile travels around a highway curve at a constant speed of 130 km/h. The constant radius of curvature of the curve is 483 m. Draw a free body diagram showing all of the forces exerted on the vehicle. Calculate the centripetal force on the vehicle as it rounds the curve. Describe clearly in words the two most probable contributions to the centripetal force.
Answer 3,320 N.
Chapter 7
Exercises
1. How was the study of the motions of projectiles useful to Newton in obtaining the gravitational force law?
2. If the earth's present mass were compressed into a sphere with a diameter only one half the earth's present diameter what would be the gravitational acceleration at the surface of this body?
A precise definition of temperature will be given in Chapter 17.
3. Estimate the acceleration due to gravity on the surface of Saturn and on the surface of Titan. Saturn's largest satellite. Calculate the acceleration of Titan in the gravitational field of Saturn.
object mass kg radius km radius of orbit km
Saturn 5.69 x 10 26 60,000 1.427 x 10 9
Titan 1.37 x 10 23 2,900 1.220 x 10 6
4. In another universe the planet Boldan describes a circular orbit about its central star Shinar. The inhabitants of Boldan know that there are three other planets in their system. To these they have given the names Alfak, Cordol, and Daphos. From their observations they discover these planets also have circular orbits around Shinar. The orbital data are given in the table below. Newton's three laws of motion hold in this universe. Discovered experimentally near the surface of Boldan all freely falling bodies have the same acceleration g. Follow the procedure in the text to obtain the gravitational force law that operates in Shinar's planetary system.
planet radius of orbit AU period years
Alfak 0.520 0.647
Boldan 1.000 1.000
Cordol 3.201 2.172
Daphos 7.300 3.763
The Boldanese have been able by geometrical means to determine Boldan is a sphere of radius R. Using their gravitational force law they have obtained the value M for the mass of Boldan. Generalize the force law obtained above to give the gravitational force between two point masses m 1 and m 2 seperated by a distance r. Express your result in terms of the given parameters m1, m2, M, g, R and r.
5. Suppose a planet is found to revolve around the sun in a circular orbit lies between the orbits of Mercury and Venus at a distance of 8.15 x 10 7.
A similar exercise appears in B.M. Casper and R.J. Noer op cit p 234.
km from the sun. The sun exerts exactly the same gravitational force on this planet that the sun exerts on the earth. What is the mass of this planet?
Answer 1.76 x 10 24 kg.
6. The centers of two identical 200 kg brass spheres are seperated by 47.3 cm. Compare the gravitational force one of the spheres exerts on the other with the gravitational force the earth exerts on one of the spheres.
Answer The earth's force is about 165 million times greater.
7. Suppose you repeat Cavendish's experiment figure 7.4 using lead balls with the smaller masses equal to 789 g and the larger ones 167 kg. These are comparable to the masses used by Cavendish. From your measurements you find when the large mass and the small mass are seperated by 21.5 cm, the value you obtain for the force between the two is 1.92 x 10 -7 N. From your data obtain a value for the gravitational constant G.
8. A astronaut weighs 175 lb on earth. What would be his weight if he stood on the surface of Mars? On Neptune?
9. Consider a object placed directly between the earth and the moon. At what distance from the center of the earth must the object be placed such the combined gravitational force exerted on it by the earth and the moon will be zero?
10. Calculate the force the sun exerts on the earth and the force the sun exerts on Jupiter. Which is greater? How many times greater?
Answer 3.53 x 10 22 N, 4.14 x 10 23 N, 11.7
11. Three small masses are arranged along a straight horizontal line as illustrated in the drawing. The horizontal gravitational force on the mass m is zero. Find the value of the mass M.
<15cm . 35 cm.
.2.73 kg .m .M
12. Uranus and Neptune describe nearly circular orbits around the sun with both orbits lying very close to the ecliptic plane. Obtain a estimate of the gravitational force that each of these planets exerts on the other when they are nearest to each other.
13. Estimate the gravitational force that Venus exerts on the earth when the two planets are closest to each other and when they are at their greatest separation.
Answer 1.1 x 10 18 N, 2.9 x 10 16 N.
14. Compare the gravitational force that Jupiter exerts on the earth at closest approach with the gravitational force that Venus exerts on the earth at closest approach.
Answer Jupiter's force is about 1.7 times larger than the force Venus exerts.
15. In a simple model of the hydrogen atom a electron describes a circular orbit about the proton. The radius of the orbit is about 5.29 x 10 -11m. Calculate the gravitational force the proton exerts on the electron in this model.
Answer 3.63 x 10 -47N.
16. The average density of a certain planet is found to be 4,730 kg/m3. The mass of the planet is 7.38 x 10 24 kg. Calculate the escape velocity for this planet.
Answer 11.7 km/s.
17. A spherically shaped asteroid with a radius of 187 km has a average density of 2,730 kg/m3. Calculate the escape velocity at the surface of this body.
18. Calculate the escape velocity for the moon and for Mercury.
Answer 2.4 km/s, 4.2 km/s.
19. The average density of a spherically symmetric planet is p. Obtain a expression for the escape velocity of this planet in terms of p. Does the escape velocity depend on any other physical parameters of this planet? If so which ones? Explain.
20. With a diameter of 940 km Ceres is the largest of the minor planets. The mass of Ceres is 1.03 x 10 21 kg. Calculate the average density of this planet. Estimate the acceleration due to gravity on the surface of Ceres. Suppose that by bending your knees and springing vertically upward your feet rise to a height of one foot above the ground. If you did this on Ceres about how high above the surface of Ceres would your feet rise? Explain clearly your reasoning.
Answer 2,370 kg/m3, 0.311 m/s2, 32 ft.
21. A 875 kg artificial satellite is in a circular orbit around the earth at a altitude of 545 km. Calculate the total energy acquired by the satellite in taking it from its position at rest on the launch pad at Cape Canaveral and placing it in this orbit
Answer 3.0 x 10 10 J.
22. In Exercise 6.15 a graphical analysis of the orbital data for the planets yields the relationship.
ac = K
T 4/3
between the planet's centripetal acceleration a and the period T of its orbit with a expressed in AU/yr2 and T in years. The constant parameter K is found from the graph to have the value K = 39.8 AU/yr 2/3. Show that these results are consistent with the centripetal acceleration a of the planets obtained from the gravitational force law.
GM sum
ac= r2
Find the relationship between the parameter K and the mass of the sun Msum. Use this value of K to obtain a value for the solar mass in kilograms.
23. Calculate the total energy kinetic and potential of the earth as it orbits the sun. Assume a circular orbit and ignore the effects of the other planets and the earth's rotation.
Answer -2.64 x 10 33 J.
24. Orbital data for the four Jovian satellites discovered by Galileo in 1609 are given in the table. Assume that the radius r of the orbit and the period T are related by a power law.
r = CT x.
where C and x are constants. Taking the logarithms of the two sides of this equation yields log r = x log T + log C.
Satellites of Jupiter.
r T
satellite km days
Io 4.22 x 10 5 1.77
Europa 6.71 x 10 5 3.55
Ganymede 10.70 x 10 5 7.16
Callisto 18.80 x 10 5 16.69
Plot log r versus log T and obtain from the graph the values of x and C. How does this relationship between r and T compare with Kepler's third law for the orbits of the planets around the sun?
25. One of the brighter stars in the sky is in the constellation Centaurus. The reason it appears so bright is due partly to the fact that it is at a distance of about 4.3 light years from the sun. A light year is the distance light travels in free space in one year. No other known star lies closer to the sun. Actually this is a system of three stars gravitationally bound together. The brightest of the three components is very similar to the sun. Calculate the distance in meters from the sun to this star system. Assume that the brightest component of the system has the same mass as the sun and calculate the gravitational force the sun exerts on it. Calculate the ratio of this force to the force the sun exerts on the earth. Calculate the ratio of the gravitational force this star exerts on the earth to the gravitational force the sun exerts on the earth. How many times greater is the gravitational force the sun exerts on the earth compared to the gravitational force the next nearest star exerts on the earth?
Answer 4.07 x 10 16 m, 1.59 x 10 17 N, 4.52 x 10 -6, 7.36 x 10 10 times.
26. How far from the surface of the sun would a object have to be for the gravitational force on it to be equal to its weight on the surface of the earth?
Answer 2.98 x 10 6 km.
27. We shall attempt to discover a power law * of the form.
rn = abn, (1)
where rn is the mean orbital radius of the nth planet counting out from the sun and a and b are constants to be determined.
actual predicted
rn rn
n planet AU log rn AU
1 Mercury 0.39
2 Venus 0.72
3 Earth 1.00
4 Mars 1.52
Jupitar 5.20
Saturn 9.54
Uranus 19.19
Neptune 30.06
Pluto 39.53
X
Y
Z
Calculate log rn for the known planets given in the table. Beginning with only the first four planets out to Mars make a plot of n versus log rn.
*L.Basano and D.W. Hughes II Nuovo Cimento C2 1979 p. 505.
With your ruler draw a straight line through the data making the best fit you can make to these four points. Keeping in mind that n must be a integer fit Jupiter and Saturn on this line as well as you can and record the corresponding values of n in the table. Next make a new plot of n versus log rn using the six known planets out to Saturn and the corresponding n-values you recorded in the table. Again draw through the data the best straight line you can. Fit Uranus, Neptune, and Pluto to this line and record in the table the n-values you obtain for these three known outer planets. Finally, make another new plot of n versus log rn using the data you now have for all nine known planets. Draw through the data the best straight line you can.
By taking the common logarithm of both sides and rearranging terms.
Eq 1 can be written as
1 log a
n = log r n 2
log b log b
which shows a linear relationship between n and log rn as does your final graph. From the slope and the vertical intercept of your graph, determine a and b. Using these values of a and b rewrite Eq. 1.
Your results should predict the existence of three missing planets denoted by X, Y, and Z in the table. Relative to the orbits of the known planets where are these missing planets? Use the equation you obtained to calculate rn for all planets known and missing ones. Enter your predicted values for rn in the table.
Calculate the semimajor axis of the object described in Exercise 1.13.
How does the object fit with the results you obtained in this exercise? Explain clearly. What is the implication of the discovery described in Exercise 1.8 with regard to your results? Explain clearly.
Chapter 8
Exercises
1. A spy satellite is placed in a circular orbit around the earth. The satellite makes fourteen complete orbits in each twenty four hour period. Calculate the altitude of this satellite.
Answer 904 km.
2. In May 1973 Skylab was launched into a nearly circular orbit 430 km above the surface of the earth. Calculate the number of orbits Skylab made around the earth each day.
Answer 15.5
3. A spacecraft in a circular orbit around the moon makes one complete orbit in exactly 2 h 10 min. How high is the spacecraft above the moon's surface?
Answer 224 km.
4. Use orbital data to determine which is larger the acceleration of the earth as it orbits the sun or the acceleration of the moon as it orbits the earth. How many times larger?
5. Space engineers wish to place a satellite in a circular orbit around the planet Mercury. They want the satellite to circle Mercury at a altitude of 150 km. Calculate the period of the orbit.
Answer 93 min.
6. A cannon on the moon's surface fires a cannonball horizontally. Calculate the speed with which the ball must leave the muzzle of the cannon in order to make a circular orbit about the moon.
Answer 1.68 km/s.
7. A communications satellite remains above the same point on the earth's surface at all times. This means that it must make exactly one complete orbit about the earth each time the earth makes a complete rotation on its axis. Calculate the approximate height of this satellite above the earth's surface.
Answer 35,880 km.
8. In 1609 with his newly constructed telescope Galileo discovered four satellites orbiting the planet Jupiter. These objects were later given the classical names Io, Europa, Ganymede, and Callisto. Ganymede is one of the largest satellites in the solar system larger in fact than the planet Mercury. It moves in a nearly circular orbit around Jupiter once in about 7.16 days at a average distance of 1.07 x 10 6 km from the center of the planet. From these data estimate the mass of Jupiter.
9. The minor planet Ceres orbits the sun with a period of 4.61 yr. Assume the orbit is circular and use Newton's gravitational force law to estimate how far Ceres is from the sun.
Answer 4.14 x 10 8 km.
10. The densities of brass and steel are 8,600 kg/m3 and 7,380 kg/m3 respectively. Two uniform metal spheres have equal diameters. One sphere is solid brass and the other is solid steel. Their centers of mass are separated by a distance of 67.3 cm. How far from the center of the brass sphere is the center of mass of this two body system?
Answer 31.1 cm.
11. The two stars in a binary system are found to describe circular orbits with a period of 26.7 yr. The radius of one orbit is seen to be 1.68 times as large as the other. The stars are 2.45 x 10 9 km apart. Calculate the masses of the components of this system.
Answer 2.30 M, 3.86 M.
12. The two stars A and B in a binary system are 8.38 AU apart as illustrated in the scale drawing below. Their orbits are circular with a period of 9.7 yr. Calculate the masses of these stars.
mB orbit of A
mA
orbit of B
inside of circle.
13. A certain nearby star is observed for several decades. Its path in the sky shows a slight wobble suggesting that it has a dimmer companion star. From the observations it is determined the orbits are circular and the radius of the orbit of one of these stars is 0.82 AU and that of the other is 2.68 AU. The orbital period of this binary system is 1.97 yr. Calculate the masses of these stars.
14. Two stars gravitationally bound to each other travel in circular orbits about the center of mass of this binary system with a period of 3.16 yr. The orbital radius of one of these stars is 1.05 AU and that of the other is 2.84 AU. Calculate the mass of each of these stars.
Answer 1.60 M, 4.34 M.
15. Two stars separated by a distance of 7.29 x 10 8 km revolve in circular orbits about a common center of mass. The center of mass of the two body system is located 2.78 x 10 8 km from the center of the heavier star which has a mass of 4.18 x 10 30 kg. How long does it take for one of the stars to make a complete orbit around the center of mass of the system?
Answer 5.84 yr.
16. The five large satellites of Uranus were discovered prior to 1950 using earth based telescopes. The presence of ten smaller satellites orbiting around Uranus was revealed in 1986 when the spacecraft Voyager II passed through the vicinity of the planet. Orbital data for each of these are given in the table. Use the data to plot b3 versus T2. From the slope of your graph estimate the mass of Uranus.
Answer 8.73 x 10 25 kg.
Orbital Data for Satellites of Uranus.
semimajor axis period
b T
satellite km hours
Cordelia 49,700 8.0
Ophelia 53,800 9.0
Bianca 59,200 10.4
Cressida 61,800 11.1
Desdemona 62,700 11.4
Juliet 64,600 11.8
Portia 66,100 12.3
Rosalind 69,900 13.4
Belinda 75,300 14.9
Puck 86,000 18.3
K.R. Lang Astrophysical Data.
17. If all of the planets in the solar system were replaced by a single star with the same mass as the sun and this star were placed at the same distance from the sun as the earth is now what would be the period of the orbit as these two stars revolve about their common center of mass?
Answer 259 days.
18. How far from the center of the earth is the center of mass of the earth moon system?
Answer 4,700 km.
19. Two planets are gravitationally bound to each other. In addition to their orbital motions about their central star they describe circular orbits around a common center of mass. The planets are assumed to have the same composition and the same average density. However the diameter of one planet is twice as large as that of the other. The distance between the centers of mass of the two planets is 609,000 km and the period of their orbits about their center of mass is 47.3 days. Calculate the mass of each planet.
Answer 8.89 x 10 23 kg and 7.12 x 10 24 kg.
20. A planet has a single natural satellite. Both have the same average density of 4,730 kg/m3. The center of mass of this two body system lies at the surface of the planet which has a radius of 8,320 km. The distance between the centers of mass of the planet and its satellite is 427,000 km. Calculate the time it takes for the planet and its satellite each to make a complete orbit about the center of mass of the system.
Answer 23.0 days.
21. The ancients discovered by geometrical means the sun is much farther away from the earth than the moon. Similar geometrical constructions based on modern observations of the moon and the sun reveal the sun is approximately 390 times as far from the earth as the moon. The orbital period of the earth has long been known to be about 365.25 days and that of the moon about 27.3 days. Using only these data and Eq. 8.5 obtain a value for the mass of the earth in units of the solar mass M. Assumw Mearth<< Msun and M moon << Mearth.
22. Sirius is the star that appears brightest in our night sky. Around the middle of the last century it was discovered that Sirius has a tiny companion which is massive enough the orbital motion of the primary brighter component can be detected. From observing Sirius over several decades it has been determined the orbital period of the system is about 50 yr and the semimajor axis of the relative orbit is about 3.0 x 10 9 km. The observations also reveal the companion is always seen to be about 2.3 times farther from the center of mass than the primary star. From these data estimate the mass of the bright star Sirius and the mass of its small dim companion.
23. Algol is a moderately bright star in the constellation of Perseus. It varies in brightness periodically because it is a binary system with the plane of the orbit oriented so that as seen from the earth one star passes in front of the other briefly blocking some of its light. From this fluctuation in brightness the period of the orbit is determined to be about 2.87 days. The semimajor axis of the relative orbit of the system is about 9,800,000 km. A analysis of the system's light spectrum reveals the mass ratio for the two components is about 5/23. Use these data to calculate the mass of each of these stars. Express your answer in terms of M.
Chapter 9
Exercises
Table 9.3. Mass and Angular
Momentum in the Solar System
angular
object mass momentum
% %
sun 99.87 0.6
outer planets 0.13 99.0
terrestrial planets <0.01 0.2
asteroids <0.01 0.1
comets <0.01 0.1
1. A straight uniform rod of length 83 cm has a mass of 486 grams. The rod is supported in the horizontal position by a string attached to one end and a finger at the other. When the supporting finger is removed what is the torque on the rod relative to a axis through the end of the rod with the string support?
Answer 1.98 Nm
2. The handle of a mechanic's wrench is 30 cm long. To tighten a bolt with this wrench the mechanic applies a force of 100 N. What is the maximum torque the mechanic can apply to the bolt using only this wrench and this force? Describe clearly where and how the force must be applied to achieve the maximum torque.
3. A wooden pin is glued into a 1.83-m wooden rod at a point 27 cm from one end of the rod. The pin is fastened to a vertical wooden post so the rod is held fixed at a angle with respect to the horizontal as shown in the scale drawing. A 250-g ball attached to a string hangs vertically from the upper end of the rod. Calculate the torque exerted on the rod by the string about a axis through the pin
Answer 3.61 Nm
4. A tapered rod has a mass of 780 grams and a length of 2.85 meters. When suspended at its midpoint the rod hangs horizontally with a 260- gram weight hanging from its narrow end. Find the position of the center of gravity of the rod.
Answer 95 cm from the thicker end.
5. A nonuniform rigid rod of mass 566 g is made to hang horizontally by means of a string attached to a point 33.6 cm from the left hand end of the rod. A 443-g weight hangs from a point on the rod 57.6 cm from this same end. How far from this end of the rod is the center of gravity of the rod located?
Answer 14.8 cm.
6. A 52 lb uniform walkboard 12 ft long is supported at its ends by two vertical walls. Find the force exerted on each wall when a 160 lb carpenter stands on the walkboard 3.5 feet from one end.
Answer 72.7 lb, 139.3 lb.
7. A 55 kg window washer is 1.45 m from one end of a uniform 35 kg platform that is 7 m long. Her water bucket weighing 220 N is 2.75 m from the same end. The platform is supported by ropes at the two ends. Calculate the tension in each rope.
Answer 370 N, 732 N.
8. After some holes are drilled in a meter stick the stick has a mass of 107 g with its center of gravity at the 527mm mark. By means of a light string a 185 g weight is attached to the stick at the 831 mm mark. At what single point on the stick must the system be supported so the stick is horizontal and in equilibrium?
Answer 720mm mark.
9. A 25.4 kg uniform beam 3.97 m long is fastened to a vertical wall at one end by a pin. The beam is held horizontal by means of a cable attached to the other end of the beam as shown in the scale drawing. Use the scale drawing to find the tension in the cable.
Answer 280 N.
Wall cable
pin beam
10. A 44.0 kg traffic light is suspended from one end of a rigid uniform metal strut of mass 6.4 kg. The other end of the strut is fastened to a vertical pole. The system is held in equilibrium by means of a cable as illustrated in the scale drawing. Use the scale drawing to calculate the tension in the cable.
Answer 420 N.
11. A rigid uniform metal beam is fastened to a vertical wall by means of a hinge at one end of the beam. The length of the beam is 3.30 m and it has a mass of 10.45 kg. A 13.92 kg light fixture is suspended from the beam. One end of a thin horizontal rod is attached to the beam. The other end of this rod is fastened to the wall. This holds the beam at a angle with respect to the wall as shown in the scale drawing. Calculate the tension in the rod and the force magnitude and direction the hinge exerts on the beam.
Answer 250 N, 350 N at 44 above the horizontal to the right.
12. A man stands at one end of a uniform plank that is 3.25 meters long. The plank is supported by a vertical post at the other end and by a heavy duty spring scale connected to the plank at a point 1.08 m from the man. The plank weighs 53 N and the scale reads 1,140 N. How much does the man weigh?
Answer 735 N.
13. Weights of 8, 5, 3, and 10 N are located respectively at the 25-, 50-, 75-, and 100-cm marks on a meter stick whose weight is negligible. What are the magnitude and location of the single upward force that will balance the system?
Answer 26 N at 64.4-cm mark.
14. A rigid metal plank of uniform density is 19.6 ft long and weighs 63.2 lb. It is supported by a slim vertical post which is located 6.75 feet from one end of the plank. A cable is fastened to the bottom of the plank as shown in the scale drawing. The other end of the cable is fastened to the floor so the cable makes a angle with the plank as shown. A 182 lb man stands 1.58 ft from the opposite end of the plank. With the plank horizontal calculate the tension in the cable.
Answer 430 N.
15. Assume circular orbits and calculate the orbital angular momentum of the earth and Jupiter relative to the sun. How many times larger is Jupiter's orbital angular momentum than the earth's?
Answer 720 times larger.
16. The mass of Pluto is only about two thousandths of the earth's mass but Pluto is very far from the sun compared to the earth. Which has the larger orbital angular momentum the earth or Pluto? How many times as large? Assume that both planets have circular orbits. This is not a very good approximation for Pluto.
17. In 1957 the Soviet Union launched the world's first artificial satellite the 84-kg Sputnik. Sputnik circled the earth every ninety six minutes at a altitude of 575 km. Calculate the orbital angular momentum of Sputnik.
Answer 4.42 x 10 12 J-s.
Chapter 10
Exercises
1. What is the force that a electric charge of one coulomb exerts on a second charge of one coulomb when the two charges are separated by a distance of one meter? Express your result in pounds. Based on your result what statement would you make about the unit of electric charge the coulomb?
2. Three point charges are arranged along a line as shown in the drawing. k - 10 cm * 15 cm * 22 cm
9 uC -4uC 7uC P
Find the magnitude and direction of the force which the 9-uC charge exerts on the 4-uC charge. Do the same for the force which the 7-uC charge exerts on the 4-uC charge. If the -4-uC charge were free to move under the influence of these forces in which direction would it move? Explain clearly and completely your reasoning. Calculate the resultant electric field at the point P due to all three charges.
Answer 32.4 N to the left, 11.2 N to the right, to the left, 1.40 x 10 6 N/C.
3. Two point charges separated by a distance of 25 cm repel each other with a force of 0.140 N. Find the force if the separation is increased to 50 cm. If one of the charges is + 1.20 uC what is the value of the other charge? Explain clearly and completely your reasoning.
Answer 0.035 N, 0.810 uC.
4. A electric charge +9.80 uC is placed to the left of a -3.20-uC charge. How far apart are these charges if the electric field at a point 20 cm to the right of the -3.20-uC charge is exactly zero? Calculate the force each charge exerts on the other when they are separated by this distance.
Answer 15cm, 12.5 N.
5. Two charged particles exert a force of 4.72 N on each other. What will be the force each will exert on the other if they are moved so they are only one seventh as far apart?
6. Assume the innermost electron in a copper atom is 1.4 x 10-12 meter from the nucleus of the atom. Calculate the magnitude of the electric force between the nucleus charge = 29e and this electron.
7. Two electric charges separated by a distance of 1.47 m exert a force of 2.72 N on each other. What will be the force each exerts on the other if one of the charges is moved so they are only 42 cm apart?
8. Calculate the work done in moving a 125-nC charge a distance of 30 cm in a unifrm electric field of 2,300 N/C. Consider the displacement to be in the direction opposite to the field.
Answer 8.625 x 10 -5J.
9. Three electric charges are arranged in a straight line as shown in the drawing. Find the electric field magnitude and direction at the point P which lies on the same line as the charges. What would be the electric force magnitude and directin on a -4-uC charge placed at the point P?
15 cm 20 cm 25 cm
9 uC -5uC p -7uC.
10. A certain point charge exerts a force of 0.054 N on a small 3-nC point charge situated 50 mm away from it. Calculate the electric field at the position of the 3-nC charge due to the first charge. What is the magnitude of the first charge?
Answer 1.80 x 10 7 N/C, 5 uC.
11. A 9-nC electric charge stands 0.138 mm to the left of a -4-nC charge. Calculate the total electric field at the point 0.217 mm to the right of the -4-nC charge.
Answer 1.22 x 10 8 N/C toward the left.
12. A point charge of 3.21 uC moves in a circular orbit of radius 62.7 cm. The charge makes five thousand complete orbits each second. Calculate the electric current represented by this moving electric charge.
Answer 16 mA.
13. Three point charges +8nC, +5 nC, and -7 nC are arranged along a line with the +5-nC charge located at a point exactly halfway between the other two. The charges on the ends are 4.24 mm apart. Calculate the electric field magnitude and direction at a point exactly halfway between the +5-nC charge and the -7-nC charge.
Answer 1.032 x 10 8 N/C toward the right.
14. For the system of charges in Exercise 13 calculate the total electric force exerted on the 8-nC charge by the other two charges.
Answer 0.0521 N toward the left.
15. Two very long parallel wires 17.3 mm apart carry the same current in opposite directions as shown. Find the magnetic field magnitude and direction at the two points P midway between the wires and P2 15.2 mm below the bottom wire.
Answer 1.51 x 10 -4 T into the page, 2.29 x 10 -5 T out of the page.
3.27 A
height is 17.3 mm P1
3.27 A
P2.
16. A proton carries a positive electric charge of magnitude e. A proton traveling to the right with a speed 2.33 x 10 6 m/s enters a uniform magnetic field 2.38 T directed toward the top of the page as shown. What is the magnitude and direction of the force on the proton as it enters the field?
Answer 8.87 x 10 -13 N out of the plane of the page.
proton > ^B
17. In the example on page 136 the magnetic force on the electron is perpendicular to the velocity. Given this fact describe the path of the electron. Explain clearly your reasoning. Calculate the acceleration of the electron. What is the radius of curvature of its path?
Answer 1.84 x 10 17 m/s2, 1.95 um
18. A electrically charged particle with linear momentum 8.32 x 10 -21 kg.m/s enters a uniform magnetic field of 0.0635 T. The direction of motion of the particle is perpendicular to the magnetic field. The path of the particle is a circular arc of radius 27.3 cm. Find the electric charge on this particle.
19. A proton enters a uniform magnetic field of strength 0.040 T and moves along a circular arc of radius 14.0 cm. Calculate the speed of this proton.
Answer 5.4 x 10 5 m/s.
20. A rectangular loop of wire lying in the plane of the page carries a steady current 1.73 A as shown. A long wire carrying a current 2.39 A lies in the same plane. Calculate the force on the current loop due to the current in the long wire.
Answer 8.93 x 10 -7 N down toward the bottom of the page.
21.7mm
1.73 A 13.3 mm
2.39 A 11.0 mm.
21. A uniform magnetic field B is directed into the plane of the page. A beam of small particles enters this region as shown. In which direction is the beam deflected by the field if the particles (a) have a positive electric charge (b) have a negative electric charge (c) are electrically neutral? Be very clear.
X X X X
particle beam X X X X
X X X X
22. A nearly uniform electric field can be created in the region between two closely spaced parallel metal plates that carry equal opposite electric charges. This is illustrated in the drawing with the upper plate positively charged and the lower one negatively charged. A small particle of mass 0.60 ug carries ten thousand extra electrons. This particle is placed in this uniform field as shown in the drawing. Calculate the electric field required for the gravitational force on the particle to be exactly balanced by the electric force.
Answer 3.68 x 10 5 N/C.
+ +
.particle
23. A electron is placed in a uniform electric field that is directed vertically downward. Suppose the electron is in equilibrium in this field with the gravitational force exactly balanced by the electric force. Calculate the magnitude of the electric field. What would be the magnitude of the electric field required to hold a proton in equilibrium in this same position? What other change would be necessary for the proton to be in equilibrium?
Answer 5.58 x 10 -11 N/C, 1.02 x 10 -7 N/C.
24. In a simple model a hydrogen atom consists of a electron and a proton separated by a distance of about 5.28 x 10 -11m. Calculate the electric potential energy of the electron in the field of the proton.
Answer -4.36 x 10 -18 J.
25. The physical law associated with the currents introduced in the conducting loops of Section 10.3 is known as Faraday's law. It is conveniently expressed in terms of a scalar quantity called magnetic flux. The magnetic field vector B is related to this quantity and is often called the magnetic flux density. Consult a introductory physics text to find the precise connection between the magnetic flux and the flux density B. In terms of magnetic flux write a general statement regarding currents induced in conducting loops by magnetic fields. Based on your general statement can you conceive of a way in which a constant magnetic field could induce a current into a closed conducting loop of constant size and shape? Explain.
26. In the experiment with the two coils described in Section 10.3 the lower plot in Fig. 10.16 shows that when the switch is opened the current induced in coil B is in the direction opposite to the direction of the current induced by closing the switch. This behavior is dictated by the principle of conservation of energy and is expressed in the form of a general rule known as Lenz's law. Look up Lenz's law in a introductory physics text and use it to explain clearly why opening the closing the switch induces currents in opposite directions in coil B.
Chapter 11
Exercises
1. Two pieces of information obtained from the scattering of helium ions on thin gold foils were
some ions were scattered at large angles.
these large angle events were very rare.
How did these observations lead Lord Rutherford to his view of a atom?
Be clear logical, and complete.
2. Explain clearly why the results of the Geiger Marsden experiments were incompatible with Thomson's model of the atom.
3. Calculate the electric force between a electron and a alpha particle separated by a distance 2 x 10 -10 m approximately the radius of a atom. Is this force attractive or repulsive? Explain.
Answer 1.15 x 10 -8 N.
4. How many times larger is the electric repulsion than the gravitational attraction between two electrons?
5. One coulomb is a substantial electric charge. Two electric charges of one coulomb each are separated by a distance of one meter. Calculate the electric force that each of these charges exerts on the other. Suppose a object at rest on the earth's surface experiences a gravitational force due to the earth equal to the force these electric charges exert on each other. What would be the mass of this object? Suppose this object is a ball of solid ice. What is the diameter of this ice ball? The density of ice is about 920 kg/m3.
Answer 9 X 10 9 N, 9.16 x 10 8 kg, 124 m.
6. Explain clearly why Lord Rutherford required the atomic nucleus to have a positive electric charge.
7. According to Rutherford's interpretation of the Geiger Marsden experiments the nucleus of the gold atom was responsible for the large deflection of the alpha particle. Use words and a drawing to explain clearly and completely how the gold nucleus was able to do this. Do not omit any relevant fact.
8. Explain clearly and completely why Lord Rutherford required the atomic nucleus to be very small.
9. When the kinetic energy of the a particle in the Geiger Marsden experiment exceeds about 5 x 10 6 eV the distribution of scattered a particles begins to deviate from classical Coulomb scattering. Rutherford interpreted this anomalous scattering as evidence that at this energy the a particle begins to penetrate the nucleus. In a head-on collision at this energy how close does the a particle charge =2e come to the center of the gold nucleus charge = 79e? Assume the gold nucleus remains stationary during the collision.
Chapter 12
Exercises
1. What innovation did Planck introduce to account for the spectral distribution of light from a hot black body? In what way was this a contradiction of classical physics?
2. Whst two noticeable changes occur in a hot black body when its temperature is increased?
Planck's constant is a very tiny quantity. In mks units it has the approximate value h = 6.63 x 10 -34 Js.
Chapter 13
Calculate the work function for this metal using the original wavelength and Eq 13.2.
KE max = hc - W
A
1.242 eV . nm -W.
0.57 eV 411 nm.
W - 2.45 eV.
W apply Eq 13.2 again.
hc
KEmax = A
1.242 eV
= - 2.45eV = 0.38 eV.
incident light of wavelength 439 nm the fastest photoelectrons leave the metal with kinetic energy 0.38 eV.
Momentum of Light
Electromagnetic radiation carries momentum.
in the formation of the dust tail of a comet. The sunlight incident on the dust in the head of a comet exerts a pressure on the dust and pushes it away from the comet head to form the dust tail.
Exercises
1. A photon has a energy of 2.50 eV. Find its frequency and its wavelength in free space.
Answer 6.04 x 10 14 Hz, 497 nm.
2. The work function of mercury is 4.53 eV. Find the longest wavelength light that can produce photoelectric emission from this element and the maximum kinetic energy of photoelectrons emitted when the mercury is irradiated with ultraviolet photons of wavelength 193 nm.
Answer 274 nm, 1.90 eV.
3. When green light falls on the surface of a piece of calcium metal electrons are found to be ejected from the metal. When the same green light shines on a piece of tin no photoelectrons are observed. Use Einstein's theory of the photoelectric effect to explain the difference in observation.
4. Blue light of wavelength 476 nm is shined on the surface of a metal in a vacuum. The most energetic photoelectrons ejected from this metal are observed to have kinetic energy 0.22 eV. Give a complete quantitative description of what is observed if the light source is replaced by another one of wavelength 514 nm. Describe quantitatively what is observed if this second light source is replaced by a third one of wavelength 537 nm.
5. Electrons are ejected from a certain metal when light is shined on the surface of the metal. The kinetic energy of the fastest electrons ejected is observed for several different colors of incident light. The data are shown in the graph. What is the value of the photoelectric work function for this metal? To what physical quantity is the slope of this graph related?
kinetic enerygy eV
.
.
1.0 .
.
0.5 .
0 0 0 0 0 0
5 x 10 14
frequency Hz
6. A human eye can detect light at very low intensity. The eye is most sensitive to yellow light of wavelength around 550 nm. Suppose electromagnetic energy light of this wavelength from a low intensity source enters the eye at a rate of 10 -16 watt about the threshold of detection for the eye. Calculate the number of photons entering the eye each second.
Answer 276.
7. Ultraviolet light of wavelength 212 nm is shined on the surface of one of the metals listed in Table 13.1. The most energetic photoelectrons are emitted with kinetic energy 1.05 eV. Which metal is it?
8. Calculate the kinetic energy of the fastest electrons ejected when violet light of wavelength 410 nm is shined on a piece of cadmium.
Answer 0.33 eV.
9. When light of a certain wavelength is shined on the clean surface of a piece of lithium metal the fastest electrons leave the metal with speed 3.08 x 10 5 m/s. Calculate the wavelength of the incident light.
Answer 472 nm.
10. From the graph estimate the value of Planck's constant h and the work function W for each of the three metals sodium, calcium, and lead.
11. The fastest photoelectrons ejected from the surface of copper when irradiated by a certain light source have kinetic energy 2.38 eV. Calculate the wavelength of the radiation. Describe the radiation. Explain.
12. Photoelectrons are ejected from a certain piece of metal when ultraviolet light of wavelength 350 nm is shined on its surface. The speed of the fastest electrons emitted is measured at 4.39 x 10 5 m/s. Calculate the work function in eV for this metal.
Answer 3.00 eV.
13. The surface of a sample of potassium is irradiated with monochromatic light. Photoelectrons are observed with energies up to 0.234 eV. What color is this light? Explain clearly your reasoning.
14. Photoelectrons are observed when ultraviolet light of wavelength 291.5 nm is shined on the clean surface of a metal bar. The kinetic energy of the electrons emitted with the greatest speed is 0.18 eV. Calculate the kinetic energy of the fastest electrons ejected when ultraviolet light of wavelength 279.0 nm is shined on this metal surface. How fast are these electrons moving as they leave the surface of the metal bar?
Answer 0.37 eV 3.6 x 10 5 m/s.
15. Calculate the energy available from one gram of matter if it could be converted completely into energy. Approximately how long could a 1,200 watt electric heater be operated continuously with this energy source?
Answer 9 x 10 13 J, 2,400 years.
16. Estimate the mass equivalent in kg of a photon of yellow light () = 550 nm. Approximately what fraction of the mass of a hydrogen atom 1.67 x 10 -27 kg does this represent?
Answer 4.0 x 10 -36 kg, 2.4 x 10 -9.
17. The speed of a five gram bullet fired from a rifle is 300 m/s. Calculate the kinetic energy of the bullet. What is the mass equivalent in kilograms of this energy? Approximately what fraction of the mass of the bullet does this represnt?
Answer 2.5 x 10 -15 kg, 5 x 10 -13.
18. Under certain conditions electrons are ejected from a metal surface when light is shined on it. This is the photoelectric effect. These electrons are ejected with a range of speeds from zero up to some maximum value. Contrary to the prediction of the classical electromagnetic theory of light measurements show the energy of the fastest of these photoelectrons does not depend on the intensity of the incident light. Explain clearly and completely in words how Einstein's theory accounts for this observation.
19. The self energy of a electron is given by Eq. 10.14 with q = Q = e = 1.60 x 10 -19 C. The so called classical radius of the electron r = re in Eq. 10.14 is obtained by setting the electron self energy equal to the rest energy of the electron. Calculate rs.
Answer 2.82 x 10 -15m.
20. A bare 500-W light source radiates energy uniformly in all directions. Calculate the intensity of the radiation at a distance of 50 cm from the source. The intensity of light at a distance r from the surface is equal to the total power radiated divided by the surface area of a sphere of radius r. Suppose the radiation falls on the clean surface of a sodium block placed 50 cm from the source. According to the classical wave theory of the photoelectric effect how long would it take for a electron sitting on the surface to absorb enough energy to escape from the sodium block? See exercise 19 for the classical radius of the electron.
Answer 159 W/m2,2.9 years.
21. When green light of wavelength 493.1 nm is shined on a certain metal surface photoelectrons are ejected from the metal. Blue light of wavelength 440.0 nm shined on this surfaces yields photoelectrons with 3.2 times the kinetic energy of the fastest electrons ejected with the green light. Calculate the photoelectric work function for this metal.
Answer 2.37 eV.
22. When light of wavelength 472.2 nm is shined on the clean surface of a certain metal the fastest electrons leave the metal with speed 3.08 x 10 5 m/s. When the light source is changed to a different wavelength electrons with speeds up to 3.97 x 10 5 m/s are observed. Calculate the wavelength of this second light source.
Answer 442.2 nm.
23. Photoelectrons with a maximum kinetic energy of 0.146 eV are ejected from a metal surface when light of wavelength A is shined on the surface. Shining light of wavelength A/3 on the same metal surface produces photoelectrons with a maximum kinetic energy of 4.527 eV. Calculate A.
Answer 567.0 nm.
24. The energy released in burning a ton of coal is about 7,300 kWh. How many joules is this? What is the mass equivalent of this energy? What is the energy equivalent of the mass of a ton of coal? What fraction of the mass of the coal is equivalent to the energy released in burning it? The mass of one ton of a substance is about 907 kg.
Answer 2.6 x 10 10 J, 2.9 x 10 -7 kg, 8.2 x 10 19 J, 3.2 x 10 -10.
25. A 3.5 g ballpoint pen is lifted from the top of a desk to a height of 10 cm above the top of the desk. What is the change in gravitational potentional energy to the pen? If this amount of energy is converted into yellow light of wavelength 583 nm how many photons would be produced? What is the mass equivalent in kilograms of this amount of energy?
Chapter 14
Excercises
1. Use the Bohr model to calculate the wavelength of the blue light emitted by hot hydrogen gas. Explain clearly and completely your reasoning.
Answer 434 nm.
2. Light from hot hydrogen gas is passed through a prism spectrometer. The wavelength of one of the violet lines in the spectrum is measured at 373.5 nm. What is the initial state quantum number n-value of the electron making this transition in the atom?
Answer 13
One astronomical unit abbreviated AU is the mean distance between the sun and the earth. 1 AU = 1.50 x 10 8 km.
3. In the Bohr model of the hydrogen atom compare the gravitational force the proton exerts on the electron with the electric force the proton exerts on the electron.
Answer The electric force is about 2 x 10 39 times larger.
4. In Bohr's model of the hydrogen atom assume the electron travels in a circular orbit around the nucleus proton. Obtain a expression for the time T required for the electron to circle the nucleus exactly once when it is in the nth Bohr orbit. Calculate T for the first orbit n=1. How fast is the electron traveling in this orbit?
Answer 1.5 x 10 -16 s, 2.2 x 10 6 m/s.
5. The boundary separating the visible and the ultraviolet parts of the electromagnetic spectrum is not sharp. Radiation that is visible as violet light to one observer may be completely invisible ultraviolet to another observer. Suppose we arbitrarily define the wavelength dividing the visible and ultraviolet portions of the spectrum to be 360 nm. Calculate the shortest wavelength of the Balmer series in hydrogen. Is it in the visible or ultraviolet region according to our definition? What general statement can you make about ultraviolet radiation from hydrogen? Explain?
6. By means of words and drawings explain clearly and in detail how green light is produced by hot hydrogen gas according to Bohr's model. Do not omit any relevant fact.
7. In a cool gas a atom of hydrogen is in its ground state. This atom absorbs a photon of a certain wavelength causing the electron to make a jump to the n = 5 state. Calculate the wavelength of the radiation absorbed by the atom. In a complete sentence describe this light.
Answer 95.0 nm.
8. The ionization energy of a hydrogen atom is the minimum energy required to remove the electron completely from the atom. Calculate the ionization energy of a hydrogen atom in the third excited state. Explain clearly your reasoning.
Answer 0.85 eV.
9. Radiation from hydrogen is observed with a wavelength of 93.8 nm. Describe this radiation. According to Bohr's model what is the initial state quantum number n value of the electron in this transition?
10. A hydrogen atom absorbs violet light of wavelength 389 nm. What are the initial and final states of the electron in this transition? Explain clearly your reasoning.
Answer 2, 8.
11. A electron in a hydrogen atom makes a transition from the ninth orbit down to the third orbit. Calculate the wavelength of the radiation emitted. Describe this radiation. Explain clearly your reasoning.
Answer 923 nm.
12. A electron excited to be the n = 4 orbit in Bohr's model of the hydrogen atom can return to the ground state n = 1 in a variety of ways emitting electromagnetic radiation in the process. List all of these and calculate the wavelength of the photon emitted in each case. Identify the radiation in each case as visible, ultraviolet, or infrared.
13. Calculate the longest wavelength of ultraviolet light emitted from hot hydrogen gas. Explain clearly in terms of Bohr's model exactly how this radiation is produced.
Answer 121.5 nm.
14. In Chapter 10 we defined the electric current as the time rate of flow of electric charge. A electric current is associated with the orbiting electron in Bohr'a model of hydrogen. Calculate this current for the first and second Bohr orbits.
Answer 1,050 UA, 262 uA.
15. The continuous spectrum of light from a incandescent source is first passed through hot hydrogen gas and then through a infrared spectrometer. Three dark lines with wavelengths 1,005 nm, 1,094 nm, and 1,282 nm are observed as part of a infrared series. Use Bohr's model of the hydrogen atom to give a clear and complete explanation of how this dark line spectrum is produced. Find the final state quantum numbers n-values for each of these transitions. Explain clearly your reasoning.
Answer 7, 6, 5.
16. A wavelength of 4,170 nm is measured for a line in the infrared emission spectrum of hydrogen. First by calculating the longest and shortest wavelengths in a series find all series to which this radiation could possibly belong. A series of emission lines is characterized by the final state n-value for the electron transition. Then identify the series by finding the initial state n value that gives the best fit to the measured line. Identify clearly the initial and final state n values for this transition. Explain clearly your reasoning.
17. In Bohr's model the centripetal force on the electron is due to the electric attraction of the nucleus. Use Newton's second law to calculate the orbital speed of the electron in the first n = 1 Bohr orbit.
Answer 2.18 x 10 6 m/s.
18. In the ultraviolet spectrum of light from hot hydrogen gas a line of wavelength 94.96 nm is observed. Use Bohr's model of the hydrogen atom to explain completely and in quantitative detail how this radiation is produced. A calculation is required. Explain clearly your reasoning. Suppose the electron started from the same initial state but made a transition corresponding to a line in the Balmer series. What would be the wavelength and color of this Balmer line? Explain clearly your reasoning.
19. A hydrogen atom in a excited state is slightly heavier than it is in the ground state. In making a transition from a higher energy state to a lower energy one the atom loses some of this excess mass in the form of energy. From Eq 13.4 the mass loss Am is found to be
Ephoton
Am = C2.
By approximately what fraction is the total mass of a hydrogen atom reduced in emitting a photon of red light? The recoil kinetic energy of the atom is negligible.
Answer 2 x 10 -9.
20. Assume the binding of the electron and the proton in the hydrogen atom is due to the gravitational attraction rather than the electric attraction of the two particles. With this assumption start with Eq 14.4 and derive the expression analogous to Eq 14.11. Evaluate the constants to obtain the expression corresponding to Eq 14.12. What would be the energy of the atom in the n = 1 orbit in this case?
Answer 2.6 x 10 -78 eV.
21. A comet first seen in September of 1987 is found to have a elliptical orbit about the sun as shown in the drawing. The comet is no longer visible. Approximately when month and year would you expect this comet to become visible again?
Answer May 2054.
4.934 x 10 9 km.
22. The head of a comet is surrounded by a huge envelope of relatively cool hydrogen gas. This envelope was discovered using instruments on board a earth orbiting satellite. Use Bohr's model of hydrogen to explain clearly and completely why this envelope could not be discovered with similar instruments on the ground.
A lot of radiation comes from the sun may cause interference.
23. A certain comet discovered in 1953 approaches the sun as close as 0.338 AU. Its elliptical orbit carries it out as far as 4.095 AU from the sun. What is the period of the orbit of this comet?
Answer 3.3 years.
24. A comet seen in 1938 has a period of 156 years. The nearest this comet came to the sun was 1.122 x 10 8 km. What is the greatest distance from the sun this comet reaches? How far from the sun does it get compared to Pluto? Write your answer in a complete sentence.
Chapter 15
Exercises
1. The elements lithium Li, sodium Na, and potassium K have similar chemical properties. Use our model of the many electron atom to explain clearly why this is so.
2. The elements Ne, Ar, and Kr are all chemically inert gases. Use the model to explain why this is so.
3. Draw energy level diagrams similar to those in Fig. 15.5 and show the electron configuration for the ground state of carbon C and of silicon Si. Use your results to explain clearly why carbon and silicon have similar chemistry.
4. Draw energy level diagrams similar to those in Fig. 15.5 and show the electron configuration for the ground state of aluminum Al and of gallium Ga. Use your results to explain clearly why aluminum and gallium have similar chemistery.
5. Compare the electric force with the gravitational force that a gold nucleus exerts on a alpha particle. Which is greater? How many times greater?
6. Draw energy level diagrams similar to those in Fig. 15.5 and show the electron configurations for carbon C and germanium Ge. Based on your results what can you say about the chemistry of these two elements? Explain.
7. Continuing the energy level diagram in Fig. 15.4 what is the maximum number of electrons that can occupy the 5 g level consistent with the exclusion principle?
8. From Fig. 15.1 we see that barium Ba has chemistry similar to strontium Sr and the chemistry of hafnium Hf is similar to that of zirconium Zr. The atomic numbers of zirconium 40 and strontium 38 differ by only two units while the difference between the atomic numbers of hafnium 72 and barium 56 is much greater. In terms of our model of the many electron atom explain clearly and completely why this is so.
9. In the ground states of atoms of the heaviest elements electrons occupy the 5 f states. Calculate the maximum number of 5 f electrons allowed in a given atom by the exclusion principle.
10. On energy level diagrams similar to those in Fig. 15.5 construct the ground state electron configuration for lithium Li and fluorine F. Based on your result and considering that a full complement of electrons in the outermost energy level corresponds to a very stable configuration give a qualitative explanation for the strong chemical bond of the LiF molecule.
Chapter 16
Exercises
Each magnitude difference is a factor of about 2.5. How many magnitudes difference do we have? The difference m regular - m sirius name of star = 1.35 - (-1.46) = 2.81 yields.
2.5 2.81 = 13.1.
1. Make a drawing representing a model of the sun. Clearly label the photosphere and the atmosphere. Make a second drawing representing Bohr's model of the hydrogen atom and showing the allowed electron orbits. Use these two drawings together to explain clearly and completely how the absorption dark line spectrum of a star like the sun is produced.
2. A tiny 100 watt light source radiates uniformly in all directions. Calculate the intensity of radiation on a small surface at a distance of 45 centimeters from the source. The direction of propagation of the light is perpendicular to the surface on which the light falls.
Answer 39 W/m2.
3. A incandescent light bulb emits about four percent of its radiated energy as visible light. What is the intensity of visible light at a distance of four meters from a bare 100 W light bulb?
Answer 0.020 W/m2.
4. The intensity of radiation from a certain star is measured at 1.9 x 10 -9 W/m2. The star's distance from the earth is 143 ly. Calculate the total power radiated by this star.
Answer 4.4 x 10 28 W.
5. If the total radiative power of the sun is 3.9 x 10 26 W estimate the amount of solar energy received by the earth each second. Treat the earth as a disk that intercepts the radiation from the sun.
Answer 1.8 x 10 17 J.
6. The intensity of radiation from the sun measured by instruments on board a spacecraft orbiting the earth at a altitude of 600 km is 1.4 kW/m2. Use this information to estimate the total radiative power of the sun.
Answer 3.9 x 10 26 W.
7. A very small 7-W source emits light in all directions uniformly. How much energy from this source is absorbed each hour by a black circular disk of diameter 3.22 mm that is 89.5 cm from the source and oriented so the plane of the disk is perpendicular to the direction of incidence of the light falling on it.
Answer 0.0204 J.
8. The earth receives energy from the sun at a rate of about 1.8 x 10 17 W. If the inhabitants of the earth were required to pay for this energy at the rate electric companies charge about 7.2 cents per kilowatt hour, what would be the approximate daily energy bill for the earth?
Answer $300,000,000,000,000.
9. The apparent magnitude of the bright star Altair is +0.77. By trigonometric parallax Altair's distance is determined to be 16.5 light years. Find the absolute magnitude of Altair.
Answer +2.2.
10. Two stars have the same absolute magnitude. One is fifteen times as far away as the other. How many times brighter does the nearer one appear to be? What is the difference in apparent magnitudes of these two stars?
Answer 225, 5.91.
11. Spica is a ordinary bright B1 type star in the constellation of Virgo. Its apparent magnitude is +0.91. From the data available estimate the distance to Spica.
Answer 80 pc.
12. Antares is a M1 type star in the constellation of Scorpius. It is about 130 pc from the sun and has a apparent magnitude of +0.92. Use these data and Fig. 16.9 to obtain a description of Antares.
13. Two main sequence stars let's call them Enlil and Pharpar are known to be at distances of 183 ly and 68 ly respectively. The apparent magnitude of Enlil is observed to be +0.54. For Pharpar it is -0.43. From the data determine which is the hotter star Enlil or Pharpar?
14. A KO star with absolute magnitude +6.0 is known to be at a distance of 8.6 pc from the sun. A B5 star 104 pc away has a absolute magnitude of -2.0. Which star appears brighter to a observer on earth? How many times as bright?
Answer 10.4.
15. Hadar and Deneb are two of the brightest stars in the sky. Data for these two stars are given in the table. From the data determine which star is intrinsically the brighter. How many times as bright?
Answer 8.4 times.
apparent distance
star magnitude pc
Deneb +1.25 500
Hadar +0.63 130
16. Two red stars P and Q are observed in the sky. P is seen to have a apparent magnitude of +16.9 at a distance of 1,522 light years from the sun. The distance to Q is 368 light years and its apparent magnitude is measured at +5.0. One of these stars is a normal star and the other is a giant. By means of calculations, determine which is the giant.
17. Shaula is a normal B1 star in Scorpius with a apparent magnitude of +1.6. Estimate the distance to Shaula.
Answer 85 pc.
18. The radiative power of a distant main sequence star is 6.2 x 10 27 W. By comparison with the sun radiative power = 3.9 x 10 26 W determine the absolute magnitude and the spectral class of this star. Explain clearly and completely your reasoning.
Answer +1.5, B6.
19. The apparent magnitude of a certain star known to be just like the sun is observed to be +10.7. Estimate the distance to this star.
Answer 170 pc.
20. The distance to a particular main sequence star in the solar neighborhood is determined by trigonometric parallax to be 11.1 ly. The intensity of radiation from this star is measured at 1.6 x 10 -10 W/m2. Compare this star with the sun to determine its absolute magnitude its apparent magnitude and its spectral class. Explain clearly your reasoning.
Answer +7.6 +5.3, KO.
21. When seen in the sky two main sequence stars appear equally bright. The spectral class of one is B4. The other is F9. Which star is farther away from us? Approximately how many times farther away?
Answer 20.
22. After the triple star system a Centauri the next nearest star to the sun is a dim red dwarf known as Barnard's star. It's apparent magnitude is +9.5. Some observational evidence exists that suggests that this star may have one or more large planets orbiting around it. Barnard's star is near enough for its distance to be determined by trigonometric parallax. It is about six light years from the sun. From the data determine the absolute magnitude of Barnard's star.
Answer +13.
23. Vega is a main sequence star with a apparent magnitude of +0.03. By trigonometric parallax it is found that Vega is 26 light years away. Merak one of the bright stars in the Big Dipper is also a main sequence star of the same spectral class as Vega. The apparent magnitude of Merak is +2.40. From these data estimate the distance to Merak. Explain clearly your reasoning.
Answer 77 ly.
24. The sun is a yellow main sequence star of spectral class G2. Calculate the apparent magnitude of the sun. Explain clearly and completely your reasoning.
Answer -27.
25. Arneb is a FO star in the constellation of Lepus. It is about 280 pc away from the sun and has a apparent magnitude of +0.70. Use these data and the H-R-diagram to obtain as complete a description of Arneb as possible.
26. Explain clearly and completely why absorption lines of singly ionized helium are prominent only in O-type stars.
27. The stars a Centauri and Denebola are both main sequence stars. Their distances have been measured to be 4.3 ly and 43 ly respectively. The apparent magnitude of a Centauri is observed to be -0.01. For Denebola the apparent magnitude is +1.2. Determine which star if either is hotter. Explain clearly and completely your reasoning in arriving at your answer.
28. What characteristics of the light from a M-type star show it is cooler than G type?
29. A comparison of the main sequence in a cluster color magnitude diagram with that in the standard H-R diagram shows the apparent magnitude of the stars on the cluster main sequence are seven magnitudes larger than the absolute magnitudes of stars in corresponding positions on the standard H-R diagram. How far away is the cluster?
30. From the spectrum of a eclipsing binary it is found that the orbits are circular with a period of 5.03 yr. From the earth the orbital plane is seen edge-on. The orbital velocity of one of these stars is determined to be 7.73 km/s and the other 17.38 km/s. Find the mass of each star.
Answer 0.93 M , 2.08 M.
31. The two stars A and B of a eclipsing binary system describe circular orbits about their common center of mass with a period of 88.2 days. The system is oriented such that the orbital plane is viewed edge-on as seen from the earth. Hydrogen lines are observed in the spectra of both components. Data for three of these lines are given in the table. The maximum and minimum observed values of the wavelengths are given for each star. From the Doppler shifts evident in the data it is clear the stars have different orbital speeds, different masses. From the data determine how fast and in which direction along the line of sight the entire system is moving relative to the earth. Determine the orbital speed of each star and the distance between the two. From your results calculate the mass of each star.
star A star B
laboratory minimum maximum minimum maximum
nm nm nm nm nm
656.273 656.202 656.532 656.332 656.402
486.133 486.080 486.325 486.177 486.228
434.047 434.000 434.218 434.086 434.132
32. The optical spectrum of the element sodium is characterized by a strong yellow line with a wavelength of about 589 nm. It is this radiation that gives the sodium vapor lamps commonly used for parking lot illumination their yellowish color. Careful laboratory measurements show that this line is actually split into two components separated by about 0.597 nm. These two lines are shown in the figure as a laboratory emission spectrum. Also shown are the sodium absorption lines observed in the spectrum of a spectroscopic binary in which only light from the brighter star can be detected. The figure shows the spectrum with the shortest minimum wavelengths observed for the two lines and the longest maximum values observed. From the data estimate how fast and in what direction the system is moving relative to the earth along the line of sight. Explain clearly and completely your reasoning.
The laboratory line appears to be moving
The minimum line appears to be moving
The maximum line appears to be moving
588.8 \ 589.0 \ 589.2 \ 589.4 / 589.6 A nm
black dark line laboratory
line minimum
line maximum
Chapter 17
Core
photosphere
hydrogen atom
atmosphere
a model of the star.
Excited hydrogen atom in stellar atmosphere
nucleus 1 2 3 4 electron at 3
656 nm photon from photosphere at circle 2.
thermometer
F C
212 100 steam point
32- -0 ice point.
Mercury column for temperature measurement.
1. In terms of a molecular picture account for the temperature and the pressure of a confined gas. Use the molecular model to explain clearly how these two quantities are related to each other.
In mks units R = 8.314 J/mole. K.
X multiply.
2. Why does the pressure of a gas enclosed in a rigid walled container increase as the temperature increases? Two reasons.
3. On average which if either moves faster in air oxygen molecules or nitrogen molecules? Explain clearly.
4. By means of a piston air at atmospheric pressure is trapped in the end of a cylinder as shown in Figure 17.2. The length of the air column in the cylinder is 83.2 mm. The piston is pushed 13.7 mm further into the cylinder. When equilibrium is reached after compression what is the pressure of the air in the cylinder?
Answer 1.21 x 10 5 N/m2.
5. A cubical metal box 50 cm in length is filled with nitrogen gas sealed and placed in a room where the temperature is 24C. Nitrogen is a diatomic molecule. A gauge on the box indicates the absolute pressure inside the box is 6.0 x 10 5 N/m2. Calculate the approximate number of nitrogen molecules inside the box. What is the mass in kilograms of the nitrogen in the box?
Answer 1.83 x 10 25 molecules, 0.851 kg.
6. Identical amounts of a certain gas which behaves like a ideal gas are heated separately. The pressure of each is measured at various temperatures. The results are shown graphically below. From the data what can you say about system X relative to system Y? Explain clearly your reasoning.
pressure X for dots is shorter than system Y.
System Y has 5 dots.
7. The amount of a given substance can be represented by the mass by the number or moles or by the number of molecules of the substance. By converting from one of these representations to another complete the table below.
substance formula mass kg number of moles number of molecules.
argon Ar 2.189 x 10 24.
flourine F2 21.772
oxygen O2 140.3
water H2O 948.2
ozone O3 3.619 x 10 25.
xenon Xe 28.7
methane CH4 7.635
8. A steel tank sits in a room where the temperature is 22C. It contains 41.8 liters of argon gas at a absolute pressure of 2.39 x 10 7 N/m2. Calculate the mass in kilograms of the gas in the tank. Argon is a monatomic molecule.
Answer 16.3 kg.
9. Pure oxygen at 0C is enclosed in a rigid metal cylinder having a volume of 12 liters. If the absolute pressure in the cylinder is 2.0 x 10 6 N/m2, how many moles of oxygen are inside the cylinder? How much pressure would the oxygen exert if the temperature were increased to 73 C?
Answer 10.6 moles, 2.53 x 10 6 N/m2.
10. Helium is pumped into a balloon until the pressure in th balloon is 1.98 x 10 6 N/m2 and the diameter of the spherical balloon is 41.3 cm. The balloon is in a room where the temperature is 14C. Later in the day the room has warmed to 31C and the diameter of the balloon is measured to be 41.7 cm. What is the pressure in the balloon then?
Answer 2.04 x 10 6 N/m2.
11. A metal cylinder contains 51.8 moles of chlorine gas Cl2. Estimate the mass of gas in the cylinder.
Answer 3.67 kg.
12. The pressure of a certain gas is measured at various temperatures. The gas behaves like a ideal gas. The data are presented in the graph at the top of the following page. From the slope of the graph determine the particle density of the gas.
Answer 2.9 x 10 25 molecules/m3.
13. A automobile tire pressure gauge reads the difference between the absolute pressure in a tire and the atmospheric pressure. What would a tire pressure gauge read when applied to a flat tire? Early in the morning when the temperature is 8 C the gauge pressure in a automobile tire is 1.85 x 10 5 N/m2. Later in the day the temperature has climbed to 22 C. What will the pressure gauge read then when applied to the same tire?
Answer 1.99 x 10 5 N/m2.
14. A steel cylinder contains 19.3 moles of the monatomic gas xenon at a pressure of 4.73 x 10 5 N/m2 and temperature of 21 C. The gas behaves like a ideal gas. The cylinder has a slow leak. Fifty days later the temperature is up to 29C but the pressure is down to 2.09 x 10 5 N/m2. How many kilograms of gas leaked out during the fifty days?
Answer 1.44 kg.
15. A ideal gas at 19.3 C occupies a volume of 64.3 liters at 5.17 x 10 5 N/m2. At a later time the pressure is measured to be 2.26 x 10 5 N/m2 at 27.6 C and the volume has expanded to 75.6 liters. Make a quantitative comparison of the initial and final states of the gas and write a sentence describing what has happened to the gas in the time between the two observations.
16. By means of a moveable piston a small amount of gas at atmospheric pressure is trapped in the end of a metal circular cylinder as illustrated in the drawing. The cylinder is in thermal equilibrium with the atmosphere in the laboratory where the temperature is 22.3 C. Calculate the final pressure of the gas if the piston is pulled 28.3 mm to the left of its initial position.
Answer 9.14 x 10 4 N/m2.
17. A metal tank is equipped with a relief valve. When the pressure of the gas in the tank exceeds 8.0 x 10 5 N/m2 the valve opens and releases gas until the pressure in the tank drops again to the value at which the valve is set. On a given day the valve starts to release gas when the temperature reaches 21 C. Later in the day the temperature is up to 26 C. What fraction of the gas escapes during this time?
Answer 1.7%.
18. The water in a cylindrical water tank is 8 m deep. The tank sits vertically on top of a tower with the bottom of the tank 17 m above the ground. A air bubble with a diameter of 1.30 mm forms on the bottom of the tank. The bubble rises toward the top of the tank. What is the diameter of the bubble just as it reaches the surface of the water?
Answer 1.57mm
19. The pressure of a quantity of methane gas CH4 at 31.7 C is 3.183 x 10 5 N/m2. Assuming that methane behaves like a ideal gas calculate the density in kg/m3 of this gas.
Answer 2.01 kg/m3.
20. A cylindrical tank 123.8 cm long has a diameter of 38.5 cm. The tank is filled with 2.67 kg of fluorine gas F2 at 20.6 C. Calculate the pressure of this gas.
Answer 1.19 x 10 6 N/m2.
21. A steel tank is installed in a laboratory where the temperature is 23 C. It contains 1.393 liters of hydrogen gas H2 at a absolute pressure of 4.72 x 10 6 N/m2. Calculate the mass of the gas in the tank. Later in the day the temperature in the laboratory has increased by seven Celsius degrees. What is the pressure of the gas then?
Answer 5.38 kg, 4.83 x 10 6 N/m2.
22. A large glass jar sits in a laboratory where the temperature is 22.8 C. To create a very good vacuum air is pumped out of the jar until the pressure is 10 -12 N/m2. Calculate the number of air molecules per unit volume remaining in the evacuated jar. Estimate the average distance between adjacent molecules in this vacuum.
Answer 245 air molecules per cubic centimeter, 0.16 cm.
23. The pressure gauge on a air compressor tank reads 6.891 x 10 5 N/m2 and the temperature is 31.3 C. Later when the temperature is 23.9 C the valve is opened briefly and some of the air is released. The pressure gauge then shows 4.121 x 10 5 N/m2. Assume that atmospheric pressure remains constant at 1.034 x 10 5 N/m2. What fraction of the air escaped while the valve was open?
Answer 33%.
24. The density of matter in the space between the galaxies is about one hydrogen atom per cubic meter at a temperature of about -270 C. Calculate the pressure of this intergalactic gas.
25. In a room where the temperature is 23.3 C a small balloon is filled with pure helium gas until the pressure reaches 2.713 x 10 5 N/m2 and the diameter of the balloon is 1.332 m. Some time later the temperature in the room has dropped to 18.1 C the diameter of the balloon has shrunk to 1.281 m and the pressure of the helium is down to 2.109 x 10 5 N/m2. Some helium leaked out between the two observations. How much? Express your answer in number of moles, in number of molecules, and in kilograms.
Answer 40.4 moles, 2.43 x 10 25 molecules, 0.16 kg.
26. A balloon contains a small amount of gas at a pressure of 3.193 x 10 5 N/m2 and a temperature of 21.8 C. Later the size of the balloon has increased by 3 percent and the pressure of the gas has risen to 3.292 x 10 5 N/m2. Find the Celsius temperature of the gas at the later time. Assume no gas leaked out.
27. The temperature in the central core of the sun is approximately 1.5 x 10 7 K. The density of hydrogen in this region is about 4.5 x 10 31 atoms/m3. Assuming the matter in the sun behaves like a ideal gas, estimate the contribution of protons hydrogen nuclei to the pressure of the gas in the solar core.
Answer 10 16 N/m 2.
Chapter 18
Exercises
1. Neutrons and protons have mass and therefore exert a attractive gravitational force on each other. Explain clearly and completely why this force cannot be the force responsible for binding neutrons and protons together to form a stable nucleus.
2. When the isotope 214 Po emits a B-particle it transform into a new element. What is the new element? What is its mass number? What is its atomic proton number? Suppose this isotope emits a particle instead of a B particle. What is the new element in this case?
3. Complete the radioactive decays.
211 Po a 190 Au B+ 138 CS B- 234 U a Uranium
131 Xe Y Xenon 40 K B- 6 He B- Helium 89 Zr B+
4. A isotope of strontium 90Sr is a fairly long lived 28 year half life radioactive byproduct of nuclear fission that has found its way into the food chain through dairy products. Write the formula for B- -decay of this isotope and calculate the energy released when it decays.
5. Calculate the energy released in the ordinary radioactive B-decay of the isotopes 20F, 32P, and 14C.
Answer 7.03 MeV, 1.71 MeV, 0.16 MeV.
6. Calculate the energy released in radioactive p- decay of each of the isotopes 3H and 23Ne.
7. Calculate the average binding energy per nucleon for 56Fe.
Answer 8.79 MeV.
8. Calculate the average binding energy per nucleon for each of the tin isotopes 112Sn, 116Sn, 120Sn, and 124Sn. Which of these isotopes is most strongly bound? Explain.
9. A sulfur atom is assembled from sixteen hydrogen atoms and sixteen neutrons. Calculate the energy released in assembly and the average binding energy per nucleon.
Answer 272 MeV, 8.49 MeV.
10. Calculate the average binding energy in MeV per particle for each of the isotopes 58Mn, 58Fe, 58Co, and 58Ni. Calculate the energy released in ordinary B-decay of 58Mn. The energy released in B+-decay of 58Co is 1.29 MeV. By means of a calculation verify this result. Explain clearly your reasoning. Note: It is important to remember that it is atomic masses of neutral atoms that we use to calculate nuclear binding energies. The number of electrons must be the same in initial and final states so that the electron contributions cancel. This is automatically achieved in B-decay not in B+-decay.
11. Calculate the average binding energy per nucleon for the uranium isotopes 235U and 238U.
Answer 7.591 MeV, 7.570 MeV.
12. The mass -39 isotope of potassium K is stable. This isotope is produced in the B-decay of a radioactive isotope in which the energy released is 0.565 MeV. Identify this radioactive isotope and calculate its mass and its average binding energy per nucleon.
13. The average binding energy per nucleon for 16N is 7.374 MeV. Calculate the atomic mass of this isotope. Use your result to calculate the energy released in radioactive B--decay of this isotope. Give the symbolic representation of this radioactive decay.
Answer 16.00610 u, 10.4 MeV.
14. In 209 Bi the average binding energy per nucleon is 7.848 MeV. Calculate the atomic mass of this isotope. From your result calculate the energy released in B--decay of 209Pb. Express this decay process symbolically.
Answer 208.98042 u, 0.61 MeV.
15. The mass of a 234Th atom is 234.04358 u. Calculate the average nuclear binding energy per nucleon for this isotope. Calculate the energy released in the radioactive alpha decay of 238U.
Answer 7.597 Mev/nucleon, 4.3 MeV.
16. The nuclei of the isotopes 23Na and 23Mg are called mirror nuclei because the proton number of each is equal to the neutron number of the other. Calculate the difference in binding energy in MeV for these two nuclei. Which is the more tightly bound system? Give a physical reason why you would expect it to be more tightly bound.
17. Calculate the energy released in radioactive B+ -decay of 39 Ca. See note on Exercise 18.10.
Answer 5.48 MeV.
18. There are five stable isotopes of the element calcium 40Ca, 42Ca, 43Ca, 44Ca, and 46Ca. Calculate the average binding energy per nucleon for each of these. Which has the most tightly bound nucleus? Explain.
19. Each of the isotopes 16 O, 32S, 75AS, 109AG, 150Sm, 197Au, and 209Bi is stable. Calculate the average binding energy per nucleon for each of these. Based on your results write a general statement regarding the relationship between the average nuclear binding energy per nucleon and the mass number for stable isotopes.
20. All isotopes of radium are radioactive. The longest lived isotope is 226Ra, a a-particle emitter with a half life of 1,620 years. Write the decay formula and calculate the energy released in the radioactive decay of this isotope.
21. Natural lithium is a mixture of two isotopes 6Li and 7Li. Calculate the natural relative abundance of each of these isotopes.
Answer 7.6% and 92.4%.
22. There are two stable isotopes of boron. One has mass 10.01294 u and the other 11.00931 u. The chemically determined mass of natural boron is 10.811 u. In a sample of natural boron what fraction of the atoms will be 10B atoms?
23. What fraction of the mass of a sample of natural chlorine is the 35Cl isotope?
Answer 74.9%.
24. Complete the following nuclear processes.
a + 90Zr > +p 12C +16 O > +y
p + 37Cl > 34S+ 7Li + 10B > 14C+
+28Si > 55Fe + 3He 2H + 123Sb > +3He
n + 99Ru > +a p + 11B > +n
12C + 28Si > 40Ca + a + 65Cu > 68Zn +
25. Calculate the energy released in each of the nuclear reactions.
3H + 14N > 16 O + n and 3He + 14N > 16 O + p.
Answer 14.48 MeV, 15.24 MeV.
26. The nucleus of a atom of 27 Al is struck by a a-particle. The products of the ensuing nuclear reaction are a proton and the nucleus of another atom. Write the complete equation for this nuclear process.
27. No element has a stable isotope with mass number equal to eight. When a proton is incident on a 7Li nucleus at rest two a-particles are produced. By how much does the combined kinetic energy of the a-particles exceed the kinetic energy of the incoming proton?
28. A nuclear reaction occurs when a 3He nucleus strikes a 88Sr target nucleus. One of the two reaction products is a proton. Write the complete equation for this nuclear reaction. The product nucleus is radioactive and decays by ordinary beta decay. Write the nuclear radioactive equation for the decay process. Calculate the energy released in the radioactive decay.
29. In the fission of 235U by slow neutrons a average energy of about 200 MeV is released in each fission. The energy released in the Hiroshima nuclear explosion in August of 1945 was about 8 x 10 13 J roughly equivalent to the explosive eenergy of 20,000 tons of TNT. Estimate the mass of 235U that fissioned in the Hiroshima bomb. If 18% of the 235U fissioned estimate the mass of 235U used in constructing the bomb.
Answer 0.97 kg, 5.4 kg.
30. Explain clearly why fission fragments nuclei produced in the fission of a heavy element are usually highly radioactive.
31. There is one stable isotope of fluorine. 19F. When a beam of slow neutrons falls on a sample of natural fluorine B- particles are detected. Write the two nuclear equations that represent the complete process of formation and decay of the radioactive nucleus. Calculate the energy released in the radioactive decay.
Answer 7.03 MeV.
32. Why does nuclear fusion require such high temperatures tens of millions of degrees? Do not omit any relevant fact in your explanation.
33. The fusion of two 2H nuclei proceeds by either of the reactions.
3He + n
2H + 2H > 3H + p
Calculate the energy released in each reaction.
Answer 3.27 MeV, 4.03 MeV.
34. Given that no stable isotope with mass number equal to five exists complete the fusion reaction.
2H + 3H > +
Calculate the energy gained from the reduction in mass in this reaction.
Chapter 19
Star 4He 12 C in core.
1H 4He in shell around core.
photosphere.
red giant.
p + e > n + v.
hydrogen cloud.
hydrogen in galaxies.
Exercises
1. Explain clearly how energy is produced in the interiors of stars like the sun. Give a clear and complete explanation of why the stellar core must reach such high temperatures tens of millions of degrees before this energy production can occur. Do not omit any relevant detail.
2. The total radiative power of the sun is estimated at 4 x 10 26 W. This energy is generated in the central core of the sun in which some of the sun's mass is converted to energy according to Einstein's mass energy relation. E= mc2.
Estimate the amount of the sun's mass in kilograms converted to energy each year. The sun is expected to continue to produce energy at this rate for another 5 billion years. Approximately what percent of the sun's present mass will be converted to energy in this time interval?
Answer 1.4 x 10 17 kg, 0.04%.
3. The conversion of hydrogen to helium in the core of a star is sometimes referred to as hydrogen burning. Explain clearly and completely how this differs from what we ordinarily mean by the burning of hydrogen gas in the earth's atmosphere.
4. The net effect of the three step process represented in Eq 19.1 in which four protons are converted to helium can be expressed as
4p > a + 2e + + 3v.
where the a-particle denotes the 4He nucleus. Add the appropriate number of electrons to both sides of this equation and use the atomic masses in Table 18.3 to estimate the energy released in this process.
Answer 24.7 MeV.
5. The 3He produced in the second step of Eq 19.1 can be converted directly to 4He through fusion with a proton in the solar core according to the reaction. 1H + 3He > 4He + e+ +v.
Calculate the maximum energy of the neutrino produced in this reaction. The equation here represents a nuclear reaction. Because we use atomic masses we must be sure to take into account the appropriate number of electrons in calculating reaction energies.
Answer 18.8 MeV.
6. Helium is the second most abundant element in the universe after hydrogen. The 4He present in the solar core can complete the conversion of the 3He produced in the second stage of Eq. 19.1 through the three step process.
3He + 4He > 7Be + y.
e- + 7Be > 7Li + v.
1H + 7Li > 4He + 4He.
The isotope 7Be has mass 7.01693 u. Calculate the energy of the neutrino produced in the second step of this process.
Answer 0.35 MeV.
7. Calculate the net kinetic energy gained per reaction in the third stage of Eq 19.1.
3He + 3He > 4He + 1 H + 1H.
Answer 12.9 MeV.
8. Why are blue main sequence stars brighter than red main sequence stars? Give a clear and complete physical explanation. Include all relevant facts.
9. Calculate the average kinetic energy per particle for particles in the core of the sun where the temperature is 1.5 x 10 7 K. What is the particle speed corresponding to this nonrelativistic kinetic energy?
10. Name the three observable phases of evolution through which a low mass star like the sun passes after it leaves the main sequence. Describe the physical characteristics of the star in each phase.
11. In white dwarf stars nuclear reactions have ceased. With no energy produced from fusion to balance the gravity why doesn't the white dwarf collapse completely under gravity?
12. Certain physical properties of a white dwarf star are responsible for the visually observable characteristics that give rise to the term white dwarf. What are these physical properties? How is each related to the corresponding visual characteristics? Be very clear.
13. Write a paragraph describing the evolution of a star like the sun. Include in your discussion the roles played by nuclear fusion, the exclusion principle, planetary nebula, gravity, the hydrogen shell, and pressure.
14. Write a paragraph explaining clearly why a heavier main sequence star eventually becomes unstable and explodes as a supernova. Include all of the key factors that lead to the instability and cause the explosion.
15. Explain clearly why the sun will never explode as a supernova.
16. What causes a star that has exhausted its fuel supply to become very dense? What stops this density increase in medium weight stars heavier than the sun?
17. With regard to their stability tell how a white dwarf and a neutron star are similar. Tell how they differ.
18. If we see a massive main sequence star what can we assume about its age relative to most stars? Explain clearly your reasoning.
19. Use words and drawings to explain clearly and completely how the 21-cm radiation from hydrogen is produced. How does this radiation differ from the radiation we normally see from hot hydrogen gas? From a atomic point of view why is it different? Why is it particularly useful to astronomers in mapping the distribution of matter in our galaxy? Be very clear. Do not omit any relevant fact.
20. Why are neutrinos particularly valuable as carriers of information about stars compared to other kinds of stellar radiations? Be clear and complete.
21. Presumably all main sequence stars emit neutrinos in abundance. Why is it that currently only measurements on neutrinos from the sun are being carried out?
22. What is the solar neutrino problem? Give a clear and complete explanation. What are the implications for astronomy?
Chapter 20
Exercises
1. Use the then new 2.5-m telescope on Mt. Wilson in the 1920s and 1930s Edwin Hubblw accumulated extensive data on the nebulae which led him to two remarkable discoveries. What were they?
2. The apparent magnitude of a certain RR Lyrae variable star in a globular cluster is observed to be +16.5. How far away is the cluster?
Answer 49,700 ly.
3. The nebula M31 visible in the constellation Andromeda is about two million light years away from us. It is a spiral galaxy similar to our own Milky Way. Assume that these two galaxies are gravitationally bound and follow circular orbits about a common center of mass. Estimate the time it takes for the Milky Way to make one complete orbit about this center of mass. Estimate the mass of the Milky Way from the number of stars assuming the average mass of a star is equal to the mass of the sun.
4. The double violet line mean wavelength 395 nm in the spectrum of singly ionized calcium is very strong in the absorption spectra of many stars. In the spectrum of a certain galaxy these lines are observed to have a mean wavelength of 398 nm. From these data estimate how far away the galaxy is?
Answer 35 Mpc.
5. In a certain galaxy one of the stars is identified as a type 1 Cepheid variable. The brightness of the star is measured over a period of several months. The data are presented in the graph below. From the data estimate the distance to the galaxy. Explain clearly and completely your reasoning.
6. On a certain day when a type I Cepheid variable star reaches maximum brightness its magnitude is measured at +14.7. Continued observation shows that its brightness diminishes reaching a minimum 12.2 days later when the magnitude is 17.2. After another 8.7 days the star is back to maximum brightness again. How far away is this star? Could it be in the Milky Way galaxy? Explain clearly your reasoning.
Answer 3 x 10 5 ly.
7. A galaxy lies 240,000,000 ly away. Its spectrum shows a redshift of 0.0173. From these data estimate the value of Hubble's constant.
Answer 70.5 km/s/Mpc.
8. A faint star in a distant galaxy is identified as a type I Cepheid variable. The time it takes for the brightness of the star to change from maximum to minimum and back to maximum is exactly 100 days. The mean apparent magnitude of this star is +26.2. The spectrum of the galaxy in which this star resides shows a redshift of 0.0056. From the data obtain a estimate of Hubble's constant.
Answer 52 km/s/Mpc.
9. Each of the photographs of the spectra of the five galaxies in Fig. 20.3 also shows for comparison portions of the spectra of hydrogen and helium. These unshifted laboratory spectral lines can be used as a reference to calibrate the phtograph and obtain the shifted wavelengths of the singly ionized calcium absorption lines in the spectra of the galaxies. The laboratory wavelengths of these hydrogen and helium lines are given below. With your ruler determine from the photograph the position of each line. From these results make a graph of actual wavelength nm versus position on the photograph mm.
line nm
388.8 nm
396.5 nm
402.6 nm
410.2 nm
412.0 nm
414.3 nm
434.0 nm
438.7 nm
447.1 nm
471.3 nm
486.1 nm
492.2 nm
501.5 nm
position nm
0
The unshifted laboratory wavelengths of the two calcium lines are 393.3 nm and 396.8 nm. Measure with your ruler the position in millimeters of each of these lines in each photograph and use the graph you obtained above to determine the actual shifted wavelength in nanometers. Use these results and Hubble's law to determine the distance to each of the five galaxies to the cluster of galaxies in which it lies.
10. A certain star is identified as a type I Cepheid variable. The avarage apparent magnitude is measured to be 8.4. In 1947 this star was seen to appear brightest successively on August 15, September 24, and November 3. From the data determine whether this star is in our galaxy. Explain clearly and completely your reasoning.
11. Charles Messier was a French astronomer who flourished in the eighteenth century. Tell what his primary interest was and explain clearly how that interest led him to accomplish a task that proved to be extremely useful to other astronomers who had nothing to do with Messier's principal interest. Describe the task and explain clearly how it was useful to the other astronomers.
Chapter 21
Exercises
1. A certain quasar has a apparent magnitude of +13.2. Hubble's law implies that it is located 900 Mpc away from our own galaxy, which has a absolute magnitude of about -21. How many times brighter than the Milky Way is this quasar? Explain clearly.
Answer 200 times.
2. The measured redshift of a certain quasar is 0.133. Assuming this is a cosmological redshift estimate the distance to this quasar.
Answer 2 billion ly.
3. Hydrogen emission lines are seen in the spectrum of a quasar. The wavelength of the green line in the Balmer series is measured at 583 nm. Estimate the distance to this quasar.
Answer 3 billion ly.
4. The apparent magnitude of a quasar is observed to be +12.1. The redshift indicates that this quasar is receding from us at about 43,200 km/s. About how many times brighter than the Milky Way is this quasar? Explain clearly your reasoning.
Answer 300 times.
5. The magnitude of a certain quasar is observed to be +11.2. The spectrum of this quasar shows hydrogen lines in the Balmer series. In the figure below these data are compared with wavelengths of the corresponding hydrogen lines measured in the laboratory. Using all of the data determine the absolute magnitude of this quasar. The absolute magnitude of our galaxy the Milky Way is about -21. How many times brighter than the Milky Way is this quasar? How many years does it take light emitted by this quasar to reach earth? Explain clearly your reasoning.
quasar
laboratory
400 500 600
wavelength nm.
5. The quasar 3C273 appears in the sky as a starlike object with a apparent magnitude of +13. The optical spectrum shown in Fig. 21.1 reveals that its spectrum is significantly shifted toward longer wavelengths. This redshift is seen when the quasar spectrum is compared with corresponding lines in the spectrum of hydrogen comparison spectrum observed in the laboratory. To calibrate the photograph use a ruler with a millimeter scale to measure the interval between the green and violet lines of hydrogen in the comparison spectrum.
scale:___ mm on the photograph = ___ nm in the laboratory.
Use this scale to determine the wavelength A of the three hydrogen lines in the quasar spectrum. Assume the redshift AA is due to the expansion of the universe and use Hubble's law to estimate the distance to this quasar. From the data estimate the absolute magnitude of 3C273. How many times brighter than the sun is 3C273? How does the brightness of 3C273 compare with that of the entire Milky Way galaxy? Give a quantitative estimate. Explain clearly your reasoning.
U.S. intelligence has not detected overt signs that North Korea is preparing to conduct a nuclear weapons test, said one U.S. defense official. But such a test would presumably be underground, so preparatory work would be difficult to detect, the official said.
There was speculation here that North Korea could carry out a nuclear test on Sept. 9, the anniversary of the formation of the Democratic People's Republic of Korea, as the country is known officially.They included Pyongyang's withdrawal from the Nuclear Non-Proliferation
Treaty, cancellation of an eight-year freeze on its plutonium-production program and assertions that reprocessing had begun on spent nuclear fuel rods — an essential step toward production of nuclear weapons.
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