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Northern Virginia CommunityCollege

PHYSICS 231

Practice Problems

Tatiana Stantcheva

November 29, 2016

Partially funded by the VCCS Paul Lee Professional DevelopmentFoundation

Contents

1 Introduction 41.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Kinematics in 1 Dimension 72.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Vectors 113.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Kinematics in Higher Dimensions 134.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Dynamics 175.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Kinetic Energy 266.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1

CONTENTS

6.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Energy 357.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8 Center of Mass 408.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Momentum 449.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10 Rotation 5110.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

11 Statics 6111.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12 Gravitation 6812.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13 Fluids 7213.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

November 29, 2016 2 University Physics I

CONTENTS

14 Oscillations 7914.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

17 Temperature and Heat 8417.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

18 Kinetic Theory 8818.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

19 First Law 9019.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

20 Second Law 9420.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.3 Section C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Specific and Latent Heats 97


Chapter 1

Introduction: Measurement,Units, Uncertainty, Estimates

1.1 Section A

1. Write the following numbers in powers of ten notation: (a) 1, 156;(b) 680; and (c) 27.60.

Ans. 1.156× 103, 6.8× 102, 2.760× 101.

2. Determine the number of significant digits in each of the followingmeasurements: (a) 214; (b) 1.60; (c) 0.03; (d) 0.0860; and (e) 860.

Ans. (a) 3, (b) 3, (c) 1, (d) 3, (e) unclear.

3. Determine the conversion factor between the following units: (a) km/hto mi/h; (b) m/s to ft/s; and (c) km/h to m/s.

Ans. (a) 0.625, (b) 3.28, (c) 0.278.

1.2 Section B

4. What is the percent uncertainty in the measurement 3.76± 0.25 m?

Ans. 6.6%

4

CHAPTER 1. INTRODUCTION

5. A student measures three time intervals in seconds: 9.22× 103 s, 8.3×104, and 0.008× 106 s. Add the three measurements.

Ans. 1.00× 105 s

6. Imagine Antarctica to be semicircular with an approximate radius of2000 km. If the average thickness of the its ice cover is 3000 m, howmany cubic centimeters of ice does it contain? Ignore the curvature ofEarth.

Ans. 2× 1022 cm3

7. A fortnight is a measure of time equal to fourteen days. How manymicrosceconds are in a fortnight?

Ans. 1.2× 1012 µ s

1.3 Section C

8. One liter of oil is spilled onto a smooth lake. If the oil spreads out uni-formly until it makes an oil slick just one molecule thick, with adjacentmolecules just touching, estimate the diameter of the oil slick. Assumeoil molecules have a diameter of 2× 10−10 m.

Ans. 3× 103 m

9. Make a rough estimate of the volume of the body of an average person(in cm3).

Ans. 6× 104 cm3

10. Estimate how many water molecules are there in an average humanbeing.

Ans. 2× 1027


CHAPTER 1. INTRODUCTION

11. During a total solar eclipse, your view of the Sun is almost exactlyreplaced by your view of the Moon. Assuming that the distance fromyou to the Sun is about 400 times the distance from you to the Moon,(a) find the ratio of the Sun’s diameter to the Moon’s. (b) What is theratio of the Sun’s volume to the Moon’s volume? (c) Position a smallcoin in your view so that it just eclipses the full Moon, and measure theangle it subtends at the eye. From this and the Earth-Moon distance(3.8× 105 km), determine the Moon’s diameter.

Ans. (a) 400; (b) 6.4× 107; (c) 3.4× 103 km.


Chapter 2

Kinematics in 1 Dimension

2.1 Section A

1. The position of a particle is given by the function

x(t) = 3t3 + 5t2 .

If time is measured in seconds, and position in meters, what is theacceleration of the particle?

Ans. a = 18t+ 10 m/s2

2. The acceleration of a particle is given by:

a = (3.0m

s4)t2 + (4.0

m

s2) .

At time t = 0, the position of the particle is xo = 2.0 m and its velocityis vo = −1.0 m

s. Find the function for (a) the velocity and (b) the

position of the particle at any given time.

(a) Ans. (1.0 ms4

)t3 + (4.0 ms2

)t− 1.0 ms

(b) Ans. (0.25 ms4

)t4 + (2.0 ms2

)t2 − 1.0 mst+ 2.0 m

3. A bird can fly 15 km/h. How long does it take to fly 75 km?

Ans. 5 h

7

CHAPTER 2. KINEMATICS IN 1 DIMENSION

4. A particle moves along the x-axis. Its position as a function of time isgiven by x = 6.0t+ 8t2, where t is in seconds and x is in meters. Whatis its acceleration as a function of time?

Ans. 16 ms2

2.2 Section B

5. A car travels from town A to town B at 40 km/h and back to town Aat 60 km/h. What is the average speed of the car for the entire trip?

Ans. 48 km/h

6. An automobile manufacturer claims that its product will, starting fromrest, travel 0.40 km in 9.0 s. What is the magnitude of the constantacceleration to do that?

Ans. 9.88 ms2

7. A sports car is advertized to be able to stop in a distance of 55 m froma speed of 100 km/h. What is its average acceleration in terms of g?

Ans. 0.71g

8. A car accelerates from 12 m/s to 21 m/s in 6.0 s. Assuming constantacceleration, how far did it travel in this time?

Ans. 99.0 m

9. A ballplayer catches a ball 3.1 s after throwing it vertically upward.With what speed did he throw it, and what height did it reach?

Ans. 15.2 m/s , 11.79 m.

10. A stone is dropped from the top of a cliff. It is seen to hit the groundbelow after 2.75 s. How high is the cliff?

Ans. 37.1 m



velc

oity (

m/s

)

time (s)

0

4

8

- 4

2 64 8 10

- 8

Figure 2.1: Velocity vs. Time (Problem 11)

11. Fig. 2.1 gives the velocity of an object as a function of time. Determine

(a) the displacement of the object as it moves from t = 0 s to t = 8 s

(b) the acceleration of the object as it moves from t = 0 s to t = 2 s.

Ans. 32 m; 4 ms2

12. A bus accelerates at 1.5 ms2

from rest to 12 s. It then travels at constantvelocity for 25 s. How far does the bus travel?

Ans. 558 m

13. The acceleration of an object is given by a = (1.2 ms2

)e−αt, where α =1.2 s−1. If the object’s velocity at t = 1.0 s is v = 4.0 m/s, determinethe velocity at t = 2.0 s.

Ans. 4.21 m/s

2.3 Section C

14. A car traveling 90 km/h is 100 m behind a truck traveling 75 km/h.How long will it take the car to reach the truck?

Ans. 24 s



15. A rocket rises vertically, from rest, with an acceleration of 3.2 m/s2

until it runs out of fuel at an altitude of 1200 m. After this point, itsacceleration is that of gravity, downward.

(a) What is the velocity of the rocket when it runs out of fuel?

(b) How long does it take to reach this point?

(c) What maximum altitude does the rocket reach?

(d) How much time (total) does it take to reach maximum altitude?

(e) With what velocity does it strike the earth?

(f) How long (total) is it in the air?

Ans. (a) 87.6 m/s; (b) 27.4 s; (c) 1592 m; (d) 36.4 s;(e) 176.6 m/s; and (f) 54.3 s.


Chapter 3

Vectors

3.1 Section A

1. Vector ~V1 is 6.0 units long and points along the negative x axis. Vector~V2 is 4.5 units long and points +45◦ to the postive axis.

(a) What are the x and y components of each vector?

(b) Determine the vector ~V = ~V1 + ~V2.

Ans. (a) (−6.0, 0) ; (3.2, 3.2) ,(b) (−2.82, 3.18) or (4.3 units, 132◦).

2. ~V is a vector 14.3 units in magnitue and points at an angle of 34.8◦

above the negative x axis. What are the x and y components of ~V ?

Ans. (−11.74, 8.16)

3. Let ~V1 = −6.0i+ 8.0j and ~V2 = 4.5i− 5.0j. Determine the magnitudeand direction of

(a) ~V1 and ~V2,

(b) ~V1 + ~V2, and

(c) ~V1 − ~V2.

Ans. (a) (10, 127◦) , (6.7,−48◦) ; (b) (3.4, 117◦) ;(c) (16.7, 129◦) .

11

CHAPTER 3. VECTORS

3.2 Section B

4. A car is driven 215 km west and then 85 km southwest. What isthe displacement of the car from the point of origin (magnitude anddirection). Draw a diagram.

Ans. (281.6 km, 192.3◦)

5. Vector ~A has a magnitude 44.0 units and makes an angle 28.0◦ withthe positive x axis. Vector ~B has a magnitude of 26.5 units and makesan angle 56.0◦ with the negative x axis. Vector ~C is along the negativey axis and has a magnitude of 31.0 units. Determine the followingvectors:

(a) ~A− ~C , (b) ~A− ~B + ~C , (c) ~A+ ~B − ~C , and (d) ~c− ~A− ~B .

Ans. (a) (64.6, 53.1◦) , (b) (62.6,−31.1◦) , (c) (77.45, 71.9◦) ,(d) (77.45, 251.9◦) .

6. Let ~A = 2i + 6j − 3k and ~B = 4i + 2j + 1k. Calculate the value of(a) ~A · ~B, and (b) ~A× ~B.

Ans. (a) 17, (b) (12,−14,−20).

7. Two vectors have magnitudes of 10 and 15 units. The angle betweenthem when they are drawn with their tails at the same point is 65◦.Determine the component of the longer vector along the line of theshorter.

Ans. 6.3 units

8. Determine the value of i · (j × k).

Ans. +1

3.3 Section C

9. Let ~S = 1i+ 2j+ 2k and ~T = 3i+ 4k. What is the angle between thesetwo vectors?

Ans. 43◦


Chapter 4

Kinematics in 2 and 3Dimensions

4.1 Section A

1. The position of a particle as a function of time is given by

~r = ((7.60 m) i+ (8.85 m) j −(

1.0m

s2

)t2k).

Determine the particle’s velocity and acceleration as a function of time.

Ans. −2tk m/s; −2k m/s2

2. A particle moves at a constant speed in a circular path with a radiusof 2.0 cm. If the particle makes four revolutions each second, what isthe magnitude of its acceleration?

Ans. 12.6 m/s2

4.2 Section B

3. An ion’s position vector is initially ~r = 5.0i−6.0j+2.0k, and 10 s laterit is ~r = −2.0i + 8.0j − 2.0k, all in meters. In unit-vector notation,what is its ~v avg during the 10 s?

Ans. (−0.7, 1.4,−0.4) m

13

CHAPTER 4. KINEMATICS IN HIGHER DIMENSIONS

4. The earth has a radius of 6380 km. (a) What is the radial accelerationof an object at the earth’s equator? (b) If a rad at the equator is greaterthan g, objects would fly off the earth’s surface and into space. Whatwould the period of the earth’s rotation have to be for this to occur?

Ans. (a) 0.034 m/s2; (b) 1.4 h

5. The radius of the earth’s orbit around the sun (assumed to be circular)is 1.50 × 108 km. Assuming the earth orbits at constant speed, whatis (a) the magnitude of the orbital velocity of the earth, and (b) thecentripetal acceleration of the earth toward the sun in m/s2?

Ans. (a) 29.9 km/s, (b) 5.95× 10−3 ms2

.

6. A projectile is thrown from the top of a building with an initial velocityof 30 m/s in the horizontal direction as shown in Fig. 4.1. If the top ofthe building is 30 m above the ground, how fast will the projectile bemoving just before it strikes the ground?

Ans. 38.6 m/s

30 m/s

30 m

Figure 4.1: A Projectile in a Horizontal Direction (Problem 6)

7. A diver running 2.1 m/s dives out horizontally from the edge of avertical cliff and reaches the water below 3.0 s later. (a) How high wasthe cliff? (b) How far from its base did the diver hit the water?

Ans. 44.1 m; 6.3 m



8. A high diver leaves the end of a 5.0-m-high diving board and strikesthe water 1.3 s later, 3.0 m beyond the end of the board. Consideringthe diver as a particle, determine (a) his initial speed, ~vo, (b) themaximum height above the water reached, and (c) the magnitude ofthe velocity ~vf with which he enters the water.

Ans. (a) 3.42 m/s; (b) 5.32 m; (c) 10.5 m/s

9. A football is kicked at ground level with a speed of 18.0 m/s at an angleof 32.0◦ to the horizontal. How much later does it hit the ground?

Ans. 1.95 s

10. You throw a ball toward a wall with a speed of 25.0 m/s and at anangle of 40.0◦ above the horizontal as shown in Fig. 4.2. The wall is22.0 m from the release point of the ball. How far above the releasepoint does the ball hit the wall?

Ans. 12.0 m

vo

22 m

Wall

ϑ

Figure 4.2: A ball thrown at a wall (Problem 10)

4.3 Section C

11. A particle leaves the origin with an initial velocity ~v = 3.00i m/s anda constant acceleration ~a = −1.00i − 0.50j m

s2. When it reaches its

maximum x coordinate, what are its (a) velocity, and (b) positionvector?

Ans. −1.5j m/s, (4.5, 2.25) m



12. A particle moves in a two-dimensional plane and the function thatdescribes its displacement is given by:

~r = (5.0 + 8.0t− 2.0t2)i+ (2.0t)j ,

where t is in seconds and x in meters. (a) What is the minimum speedof the particle? (b) At what location does this happen?

Ans. (a) 2 m/s; (b) (13i+ 4j) m

13. At what projection angle will the range of a projectile equal its maxi-mum height?

Ans. 76◦

ϑϕ

Vo

d

Figure 4.3: A projectile thrown up a hill (Problem 14)

14. A person stands at the base of a hill that is a straight incline making anangle φ with the horizontal. For a given initial speed vo, at what angleθ (to the horizontal) should objects be thrown so that the distance dthey land up the hill is as large as possible?

Ans. 12

arccos (− 1tanφ

)


Chapter 5

Dynamics

5.1 Section A

1. Only two horizontal forces act on a 3.0 kg body. One force is 9.0 N,acting due east, and the other is 8.0 N, acting 62◦ north of west. Whatis the magnitude of the body’s acceleration?

Ans. 2.9 m/s2

2. While two forces act on it, a particle is to move at the constant velocity~v = (3 m

s)i− (4 m

s)j . One of the forces is ~F1 = (2 N)i− (6 N)j. What

is the other force?

Ans. (−2, 6) N

3. If a force of 25.0 N is required to start a 6.0 kg box moving across ahorizontal concrete floor, (a) what is the coefficient of static frictionbetween the box and the floor? (b) If the 25.0 N force continues, thebox accelerates at 0.5 m

s2. What is the coefficient of kinetic friction?

Ans. 0.425; 0.37

4. Suppose that you are standing on a train accelerating at 0.20g. Whatminimum coefficient of static friction must exist between your feet andthe floor if you are not to slide?

17

CHAPTER 5. DYNAMICS

Ans. 0.20

5. A child moves with a speed of 1.50 m/s when 9.0 m from the centerof a merry-go-round. Calculate (a) the centripetal acceleration of thechild, and (b) the net horizontal force exerted on the child (m = 25 kg)

Ans. 0.25 ms2

; 6.25 N

6. A crate with mass 32.5 kg initially at rest on a warehouse floor is actedon by a net forizontal force of 140 N.

(a) What acceleration is produced?

(b) How far does the crate travel in 10.0 s?

(c) What is its speed at the end of 10.0 s?

Ans. 4.31 ms2

; 215 m; 43.1 m/s

7. A student with mass 45 kg jumps off a high diving board. Using 6.0×1024; kg for the mass of the Earth, what is the acceleration of the Earthtoward the student, as the student accelerates toward Earth with anacceleration of 9.8 m

s2?

Ans. 7.35× 10−23 ms2

5.2 Section B

8. Two blocks are connected by a cord (of negligible mass) that passesover a frictionless pulley (also of negligible mass). The arrangement isknown as Atwood’s Machine. One block has mass m1 = 1.3 kg and theother has m2 = 2.8 kg. Determine the magnitude of their accelerationand the tension in the cord.

Ans. 3.6 ms2

; 17 N

9. A 15.0-kg box is released on a 30◦ incline and accelerates down theincline at 0.30 m

s2. (a) Find the friction force impeding its motion.

(b) What is the coefficient of kinetic friction?


CHAPTER 5. DYNAMICS

Ans. 69.0 N; 0.54

10. A box slides freely down a ramp 9.0 m long inclined at 8.0◦. How longdoes it take to reach the bottom? (Assume µk = 0.06)

Ans. 4.80 s

75 kg

110 kg620 N

Figure 5.1: Two Boxes on a Horizontal Table. (Problem 11)

11. Two crates, of mass 75 kg and 110 kg, are in contact and at rest on ahorizontal surface, as shown in Fig. 5.1. A 620-N force is exerted onthe 75-kg crate. If the coefficient of kinetic friction is 0.15, calculate(a) the acceleration of the system, and (b) the force that each crateexerts on the other.

Ans. 1.88 ms2

; 369 N

FF

30°

Figure 5.2: A Box on a Horizontal Surface. (Problem 12)

12. Fig. 5.2 shows a box that slides along a horizontal frictionless surface.If F = 20 N and m = 5.0 kg,

(a) what is the acceleration of the box?

(b) If the box starts from rest, how far will it have moved when it hasa velocity of 5 m/s?

Ans. 7.46 ms2

; 1.67 m


CHAPTER 5. DYNAMICS

70°

P

Figure 5.3: A Book Pressed Against the Ceiling. (Problem 13)

13. A student, crazed with final exams, uses a force ~P of magnitude 80N to push a 5.0 kg block across the ceiling of his room, as shown inFig. 5.3. If the coefficient of kinetic friction between the block and thesurface is 0.40, what is the magnitude of the acceleration of the block?

Ans. 3.4 ms2

14. An elevator cab and its load have a combined mass of 1600 kg. Findthe tension in the supporting cable when the cab, originally movingdownward at 12 m/s, is brought to rest with constant acceleration in adistance of 42 m.

Ans. 1.8× 104 N

M

2M

30°

Figure 5.4: Two Blocks and a Pulley (Problem 15)

15. Two blocks and a pulley are arranged as shown in Fig. 5.4. The coef-ficient of kinetic friction between the block and the incline is 0.29, themass M = 5.0 kg. What is the magnitude of the acceleration of thesuspended block as it falls and what is the tension in the rope?

Ans. 4.9 ms2

; 24.54 N


CHAPTER 5. DYNAMICS

A

B

30°


16. On the diagram in Fig. 5.5, mA = 4.0 kg and mB = 2.0 kg. There isno friction between block A and the incline, the coefficient of kineticfriction between B and the horizontal plane is 0.50. Determine theacceleraiton of the blocks and the tension in the cord.

Ans. 1.63 ms2

; 13.1 N

B

A

C

D

F

T1

T2

T3

Figure 5.6: Four Suspended Disks (Problem 17)

17. Fig. 5.6 shows an arrangement in which four disks are suspended bycords. The longer, top cord loops over a frictionless pulley and pullswith a force of magnitude 98 N on the wall to which it is attached.The tension in the shorter cords is: T1 = 58.8 N, T2 = 49.0 N, andT3 = 9.8 N. What are the masses of all disks?

Ans. (a) 4.0 kg (b) 1.0 kg (c) 4.0 kg (d) 1.0 kg


CHAPTER 5. DYNAMICS


18. A block of mass m1 = 3.70 kg on a frictionless plane inlined at θ = 30.0◦

is connected by a cord over a massless, frictionless pulley to a secondblock of mass m2 = 2.30 kg hanging vertically, as shown in Fig. 5.7.Determine the magnitude of the acceleration of the hanging block, andthe tension in the cord.

Ans. 0.735 ms2

, 20.8 N

1

2

30°


19. What is the acceleration of the system shown in Fig. 5.8 if θ = 30◦,m2 = 2.0 kg, and the coefficient of kinetic friction between m1 andthe incline is 0.10? Assume that the blocks start from rest and that(a) m1 = 5.0 kg, and (b) m1 = 2.0 kg.

Ans. (a) 2.9 ms2

; (b) 2.0 ms2

.

m1

m2

F

ϑ



CHAPTER 5. DYNAMICS

20. Fig. 5.9 shows a force ~F with magnitude 12 N applied to a box of massm2 = 1.0 kg. The force is directed up a plane tilted by θ = 37◦. Thebox m2 is connected by a cord to another box m2 = 3.0 kg on the floor.The floor, the plane, and the pulley are frictionless, and the masses ofthe pulley and cord are negligible. What is the tension in the cord?

Ans. 4.6 N

21. At what minimum speed must a roller coaster be traveling when upsidedown at the top of a circle if the passengers are not to fall out? Assumea radius of curvature of 8.0 m.

Ans. 8.85 m/s

22. A 1000-kg sports car moving at 20 m/s crosses the rounded top of ahill (radius 100 m). Determine (a) the normal force on the car, (b) thenormal force on the 70-kg driver, and (c) the car speed at which thenormal force is zero.

Ans. (a) 5.8× 103 N; (b) 406 N; (c) 31.3 m/s

L

v

Figure 5.10: Tarzan Swings on a Vine (Problem 23)

23. Tarzan, who weighs 820 N, swings from a cliff at the end of a 20 mvine that hangs from a high tree limb, as shown in Fig. 5.10. Whenhe reaches the bottom, his velocity is horizontal and its magnitude is5 m/s. How much is the tension of the rope at that moment?

Ans. 9.25× 102 N


CHAPTER 5. DYNAMICS

5.3 Section C

24. A small mass m is set on the surface of a sphere. If the coefficient ofstatic friction is µs = 0.60, at what angle φ would the mass begin toslide?

Ans. 31.0◦

25. A gymnast of mass m climbs a vertical rope attached to the ceiling.You can ignore the weight of the rope. Calculate the tension in the ropeif the gymnast (a) climbs at a constant rate, (b) hangs motionless onthe rope, (c) accelerates up the rope with an acceleration of magni-tude a, and (d) slides down the rope with a downward acceleration ofmagnitude a

Ans. (a) mg; (b) mg; (c) m(a+ g); (d) m(g − a).

Figure 5.11: Monkey and a Package (Problem 26)

26. A 10-kg monkey climbs up a massless rope that runs over a frictionlesstree limb and back down to a 15 kg packages on the ground (Fig. 5.11).

(a) What is the magnitude of the least accelearation that the monkeymust have if it is to lift the package off the ground?


CHAPTER 5. DYNAMICS

(b) If after the package has been lifted, the monkey stops its climband holds onto the rope, what are the magnitude of the monkey’sacceleration and the tension in the rope?

Ans. (a) 4.9 ms2

; (b) 2.0 ms2

; (c) and 120 N

27. An electron with a speed of 1.2 × 107 m/s moves horizontally intoa region whre a constant vertical force of 4.5 × 10−16 N acts on it.The mass of the electron is 9.11 × 10−31 kg. Determine the verticaldistance the electron is deflected during the time it has moved 30 mmhorizontally.

Ans. 1.54 mm

28. A 40-kg girl and an 8.4-kg sled are on the frictionless ice of a frozenlake, 15 m apart but connected by a rope of negligible mass. The girlexerts a horizontal 5.2 N force on the rope. What are the accelerationmagnitudes of

(a) the sled, and

(b) the girl.

(c) How far from the girl’s initial position do they meet?

Ans. (a) 0.62 ms2

; (b) 0.13 ms2

; (c) 2.6 m

29. A body moves in a circle of radius 20 m with its speed given by v =3.6 + 1.5t2, with v in m/s and t in s. At t = 3.0 s, determine (a) thetangential and (b) the radial acceleration

Ans. (a) 9 ms2

; (b) 14.6 ms2

30. The terminal velocity of a raindrop (m = 3× 10−5 kg) is about 9 m/s.Assuming drag force F D = −bv, determine (a) the value of b, and(b) the time required for such a drop, starting from rest, to reach 53percent of terminal velocity.

Ans. (a) 3.27× 10−5 kg/s; (b) 0.9 s


Chapter 6

Kinetic Energy

6.1 Section A

1. A proton (m = 1.67 × 10−27kg) is being accelerated along a straightline at 3.6× 1015 m/s2 in a machine. If the proton has an initial speedof 3.4 × 107m/s and travels 3.5 cm, what then is (a) its speed, and(b) the increase in its kinetic energy?

Ans. (a) 3.75× 106 m/s; (b) 2.1× 10−13 J

2. A coin slides over a frictionless plane and across and xy coordinatesystem from the origin to a point with xy coordinates (3.0, 4.0) m whilea constant force acts on it. The force has a magnitude 2.0 N and isdirected at a counterclockwise angle of 100o from the positive directionof the x axis. How much work is done by the force on the coin duringthe displacement?

Ans. 6.8 J

3. A 1300-N crate rests on the floor. How much work is required to moveit at constant speed

(a) 4.0 m along the floor against a friction force of 230 N?

(b) 4.0 m vertically?

Ans. (a) 9.2× 102 J; (b) 5.2× 103 J

26

CHAPTER 6. KINETIC ENERGY

4. How much work is required to stop an electron (m = 9.11× 10−31 kg)which is moving with a speed of 1.90× 106 m/s?

Ans. −1.6× 10−18 J

6.2 Section B

θ

A

B

P

Figure 6.1: Block Sliding on an Incline (Problem 5)

5. (0.8) A 2.0-kg block slides down a frictionless incline (θ = 20o) from p.Ato p.B. A force with magnitude P = 3.0 N acts on the block betweenp.A and p.B as shown in Fig. 6.1. Points A and B are 2.0 m apart. Ifthe kinetic energy of the block at A is 10 J, what is the kinetic energyof the block at B?

Ans. 29 J

6. (0.8) During the time a 2.0-kg projectile moves from its initial positionto a point that is displaced 20 m horizontally and 15 m above its initialposition, how much work is done by the gravitational force on theprojectile?

Ans. −0.29 kJ

7. A luge and its rider, with a total mass of 85 kg, emerge from a downhilltrack onto a horizontal straight track with an initial speed 0f 37 m/s.If a force slows them to a stop at a constant rate of 2.0 m

s2,

(a) what magnitude F is required for the force?

(b) what distance d do they travel while slowing,



(c) what work W is done on them by the force?

(d) What are the F , d, and W if they slow at 4.0 ms2

, instead?

Ans. (a) 1.7× 102 N; (b) 3.4× 102 m; (c) −5.8× 104 J;(d) 3.4× 102 N, 1.7× 102 m , −5.8× 104 J.

8. (0.8) As a 2.0-kg object moves along the x axis,the only force acting onit is given by Fx = (3x2 + 2x)N, where x is in m. If the object movesfrom x = 1.0 m to x = 3.0 m, what is its change in kinetic energy?

Ans. 34 J

9. A 285-kg load is lifted 22.0 m vertically with an acceleration a = 0.16gby a single cable. Determine

(a) the tension in the cable,

(b) the net work done on the load,

(c) the work done by the cable on the load

(d) the work done by gravity on the load

(e) the final speed of the load assuming it started from rest.

Ans. (a) 3.24× 103 N; (b) 9.83× 103 J; (c) 7.13× 104 J;(d) −6.14× 104 J; and (e) 8.31 m/s.

10. A cord is used to vertically lower an initially stationary block of massM at a constant downward acceleration of g

4. When the block has fallen

a distance d, find

(a) the work done by the cord’s force on the block.

(b) the work done by the gravitational force on the block.

(c) the kinetic energy of the block,

(d) the speed of the block.

Ans. (a) −34Mgd; (b) Mgd; (c) 1

4Mgd; (d)

√gd2

.



θ

F3

F2F1

d

Figure 6.2: Block and Three Forces (Problem 11)

11. The diagram in Fig. 6.2 shows three forces applied to a trunk thatmoves leftward by 3.00 m. The forces are F1 = 5.00 N, F2 = 9.00 N,and F3 = 3.00 N, and the angle θ is 60o. During the displacement,what is the net work done on the trunk by the three forces?

Ans. 1.5 J

12. Fig. 6.3 gives the spring force Fx in Newtons versus the position x incm for the spring-block arrangement on he diagram. We pull the blockout to x = 12 cm and then release it. How much work does the springdo on the block when the block moves from xi = +8.0 cm to

(a) x = +5.0 cm

(b) x = −5.0 cm

(c) x = −8.0 cm

(d) x = −10.0 cm

Ans. (a) 15.6 J; (b) 15.6 J; (c) 0 J; and (d) −14.4 J.

13. The force on an ideal spring exerts on an object is given by Fx = −kx,where x measures the displacement of the object from its equilibrium



x (cm)

Fx (N)

80

160

- 80

- 160

- 1- 2 1 2

Figure 6.3: Force versus Displacement (Problem 12)

(x = 0) position. If k = 60 N/m, how much work is doen by this forceas the object moves from x = −20 cm to x = 0?

Ans. +1.2 J

14. The force on a particle is directed along an x axis and given by f =Fo(

xxo− 1). Find the work done by the force in moving the particle

from x = 0 t x = 2xo.

Ans. 0 J

15. Fig. 6.4 gives the acceleration of a 2.00 kg particle as an applied force~Fa moves it from rest along an x axis from x = 0 to x = 9.0 m. Howmuch work has the force done on the particle when the particle reaches

(a) x = 4.0 m

(b) x = 7.0 m

(c) x = 9.0 m

(d) What is the particle’s speed when it reaches x = 4.0,x = 7.0, andx = 9.0 m.



6

- 6

x (m)

a (m/s2

)

3 6 9

0

Figure 6.4: Acceleration versus Displacement (Problem 15)

4 8

+ 20

0

-20

12

F (N)

x (m)

Figure 6.5: Force versus Position (Problem 16)

Ans. (a) 42 J; (b) 30 J; (c) 12 J; (d) 6.5 m/s, 5.6 m/s, and3.5 m/s.

16. (0.8) An object moving along the x axis is acted upon by a force Fxthat varies with position as shown in Fig. 6.5. What work is done bythis force as the object moves from = 2.0 m to x = 8.0 m?

Ans. +30 J

17. A 100-kg block is pulled at a constant speed of 5.0 m/s across a horizon-tal floor by an applied force of 122 N directed 37◦ above the horizontal.What is the rate at which the force does work on the block?



Ans. 4.9× 102 W

Figure 6.6: Block falling onto a spring. (Problem 18)

18. A 250-g block is dropped onto a relaxed vertical spring that has a springconstant of k = 2.5 N/cm ( 6.6). The block becomes attached to thespring and compresses the spring 12 cm before momentarily stopping.Neglect friction. While the spring is being compressed, what work isdone on the block by

(a) the gravitational force,

(b) and the spring force?

(c) What is the speed of the block just before it hits the spring?

(d) If the speed at impact is doubled, what is the maximum compres-sion of the spring?

Ans. (a) 0.29 J; (b) −1.8 J; (c) 3.471 m/s; (d) 0.23 m.

m

F

Figure 6.7: Two pulleys and a block. (Problem 19)

19. In the Fig. 6.7, a cord runs around two massless, frictionless pulleys.A canister with mass m = 20 kg hangs from one pulley and you exerta force ~F on the free end of the cord.



(a) What must be the magnitude of ~F if you are to lift the canisterat a constant speed?

(b) To lift the canister by 2.0 cm, how far must you pull the free endof the cord?

(c) During that lift, what is the work done on the canister by yourforce (via the cord) and the gravitational force?

Ans. (a) 98 N (b) 4 cm (c) 3.92 J, −3.92 J

20. How long will it take a 1750-W motor to lift a 315-kg piano to a sixth-story window 16.0 m above?

Ans. 28.2 s

21. (0.8) A 3.0-kg block is on a horizontal surface. The block is at restwhen at t = 0. A force with a magnitude P = 12 N acting parallelto the surface is applied to the block causing it to accelerate. Thecoefficient of kinetic friction between the block and the surface is 0.2.At what rate is the force ~P doing work on the block at t = 2.0 s?

Ans. 49 W

6.3 Section C

θ

A

B

Figure 6.8: Block on an Incline. (Problem 22)

22. (0.4) A 1.4-kg block is pushed up a frictionless 14o incline from p.A top.B by a force with magnitude P = 6.0 N as shown in Fig. 6.8. PointsA and B are 1.2 m apart. If the kinetic energies of the block A and Bare 3.0 J and 4.0 J, respectively, how much work is done on the blockby the force P between A and B?



Ans. 5.0 J

23. (0.5) The only force acting on a 2.0-kg body as it moves along the xaxis is given by Fx = (12− 2x) N, where x is in m. The velocit of thebody at x = 2.0 m is 5.5i m/s. What is the maximum kinetic energyatained by the body?

Ans. 46 J


Chapter 7

Potential Energy

7.1 Section A

0 m

20 m

40 m

p.A

p.B

p. C

p. D

Figure 7.1: Roller Coaster (Problem 1)

1. A roller coaster starts at rest from p.A as shown in Fig. 7.1. Assumingno friction and using the scale given in the diagram, calculate the speedat points B, C, and D.

Ans. 24.25 m/s, 9.9 m/s, and 19.8 m/s

2. A 1200-kg car rolling on a horizontal surface has speed v = 65 km/hwhen it strikes a horizontal coiled spring and is brought to rest in adistance of 2.2 m. What is the spring stiffness?

Ans. 8.1× 104 N/m

35

CHAPTER 7. ENERGY

3. If U = 3x2 + 2xy+ 4y2z, what is the function for the conservative forceassociated with it?

Ans. (−6x− 2y)i+ (−2x− 8yz)j + (−4y2)k

4. The potential energy associated with a conservative force is given asa function of x only as: U = e−3x + 3x2. Determine the conservativeforce associated with that potential energy.

Ans. 3e−3x − 6x

7.2 Section B

5. A particular force obeys the force law ~F = (−kx + ax3 + bx4)i. Showthat it is a conservative force, and determine the function for the po-tential energy.

Ans. kx2

2− ax4

4− bx5

5

6. A 0.40-kg ball is thrown with a speed of 12 m/s at an angle of 33◦.Determine what is its speed at its highest point, and how high it goes?

Ans. 10.06 m/s, 2.2 m

7. A 12-kg projectile is launched with an initial vertical speed of 20 m/s.It rises to a maximum height of 18 m above the launch point. Howmuch work is done by the dissipative resistive force on the projectileduring this ascent?

Ans. −0.28 kJ

8. (0.8) A 2.0-kg mass is projected from the edge of the top of a 20-mtall building with a velocity of 24 m/s at some unknown angle abovethe horizontal. Disregard air resistance and assume the ground is level.What is the kinetic energy of the mass just before it srikes the ground?

Ans. 0.97 kJ


CHAPTER 7. ENERGY

R

p. A

p. B

·

Figure 7.2: Mass projected down a circular track (Problem 9)

9. A 1.2-kg mass is projected down a rough circular track (r = 2.0 m) asshown in Fig. 7.2. The speed of the mass at p.A is 3.2 m/s, and at p.Bit is 6.0 m/s. How much work is done on the mass between A and Bby the force of friction?

Ans. −8.1 J

10. A 15-g ball is shot from a spring gun whose spring has a force constantof 600 N/m. If the gun is aimed vertically and the spring is compressed5 cm, how high will the ball go?

Ans. 5.10 m

11. A 75-kg trampoline artist jumps vertically upward from the top of aplatform with a speed of 5.0 m/s.

(a) How fast is he going as he lands on the trampoline, 2 m below?

(b) If the trampoline behaves like a spring with spring stiffness con-stant 5.2× 104 N/m, how far does he depress it?

Ans. (a) 8.0 m/s; (b) 0.32 m.

12. (0.8) A block (m = 4.0 kg) sliding on a horizontal frictionless surfaceis attached to one end of a horizontal spring (k = 100 N/m) whichhas its other end fixed. If the maximum distance the block slides fromequilibrium position is equal to 20 cm, what is the speed of the block atan instant when it is a distance of 16 cm from the equilibrium position?


CHAPTER 7. ENERGY

Ans. 60 cm/s

13. (0.8) As a 2.0-kg block sliding on a frictionless horizontal surface isattached to one end of a horizontal spring (k = 600 N/m) which hasits other end fixed. The speed of the block when the spring is extended20 cm is equal to 3.0 m/s. What is the maximum speed of this blockas it oscillates?

Ans. 4.6 m/s

14. A 62-kg bungee jumper jumps from a bridge. He is tied to a bungeecord whose unstretched length is 12 m, and falls a total of 31 m. As-suming Hooke’s law applies to the rope, calculate (a) the spring stiffnessconstant k of the bungee cord, and (b) the maximum acceleration thejumper experiences.

Ans. (a) 1.0× 102 N/m; (b) 22.2 ms2

7.3 Section C

Figure 7.3: Mass fastened to a spring

15. (0.6) A 20-kg mass is fastened to a light spring (k = 380 N/m) thatpasses over a pulley as shown in Fig. 7.3. The pulley is frictionless, andthe mass is released from rest when the spring is unstretched. Afterthe mass has dropped 0.40 m, what is its speed?

Ans. A. 2.2 m/s


CHAPTER 7. ENERGY

16. The nuclear force between two neutrons in a nucleus is described roughlyby the Yukawa potential U(r) = −Uo

(ror

)e−

ror , where r is the distance

between the neutrons and U0 and r0 ≈ 10−15 m are constants. Consid-ering the Yukawa Potential to be the potential energy associated withit, determine the nuclear force.

Ans. −Uo ror2

e−ror

(1− ro

r

)


Chapter 8

Center of Mass

8.1 Group A

1. Calculate the CM of the Earth-Moon system. The mass of Earth is5.98× 1024 kg; the mass of Moon is 7.35× 1022 kg; the distance Earth-Moon is 380,000 km.

Ans. 4.6× 103 km from Earth’s center

2. The distance between a carbon atom (m = 12u) and an oxygen atom(m = 16u) in the CO molecule is 1.13 A. How far from the carbonatom is the center of mass of the molecule?

Ans. 0.646A

8.2 Section B

0.5 m 0.25 m

1.00 kg 1.50 kg 1.10 kg

Figure 8.1: Three particles along a line

3. Find the center of mass of the three-particle-system shown in Fig. 8.1.(A. 0.44 m to the right of the 1.0-kg particle)

40

CHAPTER 8. CENTER OF MASS

2 4

2

0

m1

m2

m3

x (m)

y (m)

Figure 8.2: Three particles (Problem 4)

4. Figure 8.2 shows a three-particle system, with masses m1 = 3.0 kg,m2 = 4.0 kg and m3 = 8.0 kg. What are the x and the y coordinate ofthe center of mass?

Ans. xcm = 1.1 m, ycm = 1.3 m

3M

M

LL

x

y

Figure 8.3: Inverted U-Shape (Problem 5)

5. In the diagram shown in Fig. 8.3, three uniform rods with negligiblethickness, each of length L = 22 cm, form an inverted U. The verticalrods each have a mass of M = 14 g, the horizontal rod has a mass of3M = 42 g. What are the x and the y coordinates of the CM of theobject?

Ans. (11,−4.4) cm



x

Lo

2Lo

3Lo

y

Figure 8.4: Three Cubes (Problem 6)

6. Three uniform cubes with sides L0, 2L0, and 3L0, respectively arealigned symmetrically along the x axis as shown in Fig. 8.4. Assumingthe cubes are made of the same material, determine the Center of Massof the system.

Ans. (103Lo, 0)

8.3 Section C

R R/2

Figure 8.5: Circular Part (Problem 7)

7. A uniform thin machine part is a flat circular plate of radius 2R thathas a circular hole of radius R cut out of it. The center of the hole isa distance 0.80R from the center of the plate. What is the position ofthe center of mass of the plate?

Ans. (−0.027R, 0)

8. Determine the CM of a machine part that is a uniform cone of heighth and radius R.



Ans. 34h

9. A rod with length L has linear mass density given by λ = ae−bx, wherea and b are constants. Determine the location of its center of mass.

Ans.


Chapter 9

Linear Momentum

9.1 Section A

1. A 0.145-kg baseball pitched at 39.0 m/s is hit on a horizontal linedrive straight back toward the pitcher at 52.0 m/s. If the contact timebetween bat and ball is 3.00 × 10−3 s, calculate the magnitude of theaverage force between the ball and bat during contact.

Ans. 4.4× 103 N

2. A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. Thegolf club was in contact with the ball for 3.5× 10−3 s. Find

(a) the impulse imparted on the ball,

(b) the average force exerted on the ball by the golf club.

Ans. (a) 2.0 kg.m/s (b) 5.8× 102 N

3. A 95-kg fullback is running at 4.0 m/s to the east and is stopped in0.75 s by head-on tackle by a tackler running due west. Calculate

(a) the impulse exerted on the fullback.

(b) the average force exerted on the tackler.

Ans. (a) −380 kg.m/s (b) 507 N

44

CHAPTER 9. MOMENTUM

4. The force on a particle of mass m is given by ~F = 26i−12t2j, where ~Fis in N and t in s. What will be the change in the particle’s momentumbetween t = 1.0 s and t = 2.0 s?

Ans. (26.0i− 28j) N.s

5. A 12-g bullet traveling 190 m/s penetrates a 2.0-kg stationary blockof wood and emerges going 150 m/s. If the block is stationary on africtionless surface when hit, how fast does it move after the bulletemerges?

Ans. 0.24 m/s

6. A 12, 600-kg railroad car travels alone on a level frictionless track with aconstant speed of 18.0 m/s. A 5350-kg load, initially at rest, is droppedonto the car. What will be the car’s new speed?

Ans. A. 12.6 m/s

7. A 9700-kg boxcar traveling at 18 m/s strikes a second boxcar at rest.The two stick together and move off with a speed of 4.0 m/s. What isthe mass of the second car?

Ans. 3.4× 104 kg

8. A 9300-kg boxcar traveling at 15.0 m/s strikes a second 14,000-kg box-car at rest. The two stick together. Determine the velocity with whichthey move after the collision.

Ans. 6.0 m/s

9. [Note1 ]Calculate the force exerted on a rocket, given that the propellinggases are expelled at a rate of 1500 kg/s with a speed of 4.0× 104 m/s(at the moment of take off.

Ans. 6.0× 107 N upward

1Skip this problem if topic is not covered in class yet


CHAPTER 9. MOMENTUM

9.2 Section B

10. The force on a bullet is given by the formula: F = (780− 2.6× 105t) Nover the time interval t = 0 to t = 3.0 × 10−3 s. If the bullet achievesa speed of 300 m/s as a resulf of the impulse of this force, exerted onit by the barrel of a gun, what must be the bullet’s mass?

Ans. 3.9 g

x

y

Figure 9.1: A tennis ball (Problem 11)

11. A tennis ball of mass m = 0.060 kg and speed of v = 25 m/s strikes awall at 45◦ angle and rebounds with the same speed at 45◦, as shownin Fig. 9.1. What is the impulse given to the ball.

Ans. 2.1 kg.m/s, away from the wall

12. A gangster sprays Superman’s chest with 3-g bullets at the rate of 100bullets/min, and the speed of each bullet is 500 m/s. Assuming thebullets rebound straight back with no change in speed, what is themagnitude of the average force on Superman’s chest from the streamof bullets?

Ans. 5 N

13. A space vehicle is traveling at 4300 km/h relative to Earth when theexhausted rocket motor is disengaged and sent backward with a speedof 82 km/h relative to the command module. The mass of the motor isfour times the mass of the module. What is the speed of the commandmodule relative to Earth just after the separation?

Ans. 4.6× 103 km/h


CHAPTER 9. MOMENTUM

14. An atomic nucleus at rest decays radioactively into an α particle and asmaller nucleus. What will be the speed of this recoiling nucleus if thespeed of the α particle is 3.8× 105 m/s. Assume the recoiling nucleushas a mass 57 times greater than that of the α particle.

Ans. 6.7× 103 m/s

15. A rocket of mass m traveling with speed v◦ along the x axis suddenlyshoots out one-third its mass parallel to the y axis (as seen by anobserver at rest) with speed 2v◦. Express the velocity of the remainderof the object in unit vector notation.

Ans. (32v◦i− v◦j)

16. A 4.0-kg object sliding on frictionless surface explodes into two 2.0-kgparts, one moving at 3.0 m/s due north, and the other at 5.0 m/s 30◦

north of east. What was the original speed of the object?

Ans. 3.5 m/s

Block 1 Block 2

Figure 9.2: A Bullet and Two Blocks (Problem 17)

17. A 3.50-g bullet is fired horizontally at two blocks at rest on a frictionlesstable, as shown in Fig. 9.2. The bullet passes through block 1 (m = 1.20kg) and embeds itself in block 2 (m = 1.80 kg). The blocks end upwith speeds v1 = 0.630 m/s and v2 = 1.40 m/s. Neglecting the materialremoved from block 1 by the bullet, find the speed of the bullet justbefore and after block 1.

Ans. 937 m/s, 721 m/s.

18. A ball of mass 0.540 kg moving east with a speed of 3.90 m/s collideshead-on with a 0.320-kg ball at rest. If the collision is perfectly elastic,what will be the speed and direction of each ball after the collision?


CHAPTER 9. MOMENTUM

Ans. 0.998 m/s, 4.89 m/s

19. A 0.060-kg tennis ball, moving with a speed of 7.50 m/s, has a head-oncollision, with a 0.090-kg ball initially moving away from it at a speedof 3.00 m/s. Assuming a perfectly elastic collisions, what is the speedand direction of each ball after the collision?

Ans. 2.10 m/s, 6.6 m/s

20. A ball of mass 0.440 kg moving east (+x direction) with a speed of3.30 m/s collides head-on with a 0.220-kg ball at rest. If the collisionis perfectly elastic, what will be the speed and direction of each ballafter the collision?

Ans. 1.10 m/s, 4.4 m/s

21. A collision occurs between a 2.00-kg particle traveling with velocity~v1 = (−4.00 m/s)i + (−5.00 m/s)j and a 4.00-kg particle travelingwith velocity ~v2 = (−6.00)i+ (−2.00)j m/s. The collision connects thetwo particles. What then is their velocity in (a) unit-vector notation,and (b) as a magnitude and angle?

Ans. (a) (−5.33 m/s ,−3.00 m/s) (b) (6.12 m/s , 209◦)

22. A tennis ball leaves the racket of a top player on the serve with a speedof 65.0 m/s. If the ball’s mass is 60 g and is in contact with the racketfor 30 ms, what is the average force on the ball? Would this force belarge enough to lift a 60-kg person?

Ans. 130 N, no

23. A 14.0-g bullet strikes a 380-g block initially hanging at rest (ballisticpendulum). Determine what fraction of the initial kinetic energy is lostin the collision?

Ans. 96.4 %


CHAPTER 9. MOMENTUM

24. A rifle bullet with mass 8.00 g strikes and embeds itself in a blockwith mass 0.992 kg that rests on a frictionless, horizontal surface andis attached to a coil spring. The impact compresses the spring 15.0 cm.Calibration of the spring shows that a force of 0.750 N is required tocompress the spring 0.250 cm. (a) Find the magnitude of the block’svelocity just after impact. (b) What was the initial speed of the bullet?

Ans. (a) 2.60 m/s (b) 325 m/s

9.3 Section C

Figure 9.3: A steel ball collides with a block. (Problem 25)

25. A steel ball of mass 0.500 kg is fastened to a cord that is 70.04 cmlong and fixed at the far end as shown in Fig. 9.3 The ball is thenreleased when the cord is horizontal. At the bottom of its path, theball strikes a 2.50 kg steel block intiially at rest on a frictionless surface.The collision is elastic. Find the speed of the ball and the speed of theblock, both just after the collision.

Ans. −2.47 m/s, 1.23 m/s

26. A 5.20-g bullet moving at 672 m/s strikes a 700-g wooden block atrest on a frictionless surface. The bullet emerges, traveling in the samedirection with its speed reduced to 428 m/s.

(a) What is the resulting speed of the block?

(b) What is the speed of the bullet-block center of mass?

Ans. 1.81 m/s; 4.96 m/s


CHAPTER 9. MOMENTUM

27. A stone is dropped at t = 0. A second stone, with twice the mass ofthe first, is dropped from the same point at t = 100 ms. Assume bothstones are still in air.

(a) How far below the release point is the center of mass of the twostones at t = 300 ms?

(b) How fast is the center of mass of the two-stone system moving atthat time?

Ans. (a) 0.28 m (b) 2.3 m/s

28. A shell is shot with an initial velocity v◦ = 20 m/s, at an angle ofθ = 60◦ with the horizontal. At the top of the trajectory, the shellexplodes into two fragments of equal mass. One fragment, whose speedimmediately after the explosion is zero, falls vertically. How far fromthe gun goes the other fragment land, assuming that the terrain is leveland that air drag is negligible?

Ans. 53 m

29. A rocket that is in deep space and initially at rest realtive to an inertialrefrence fram has a mass of 2.55 × 105 kg, of which 1.81 × 105 kg isfuel. The rocket engine is then fired for 250 s while fuels is consumedat the rate of 480 kg/s. The speed of the exhaust products relative tothe rocket is 3.27 km/s. What is (a) the rocket’s thrust, (b) the mass,and (c) the speed of the rocket?

Ans. (a) 1.57× 106 N; (b) 1.35× 105 kg; (c) 2.08× 103 m/s.

30. A 2.0-kg block slides along a frictionless tabletop at 8.0 m/s towarda second block (at rest) of mass 4.5 kg. A coil spring, which obeysHooke’s law and has spring constant k = 850 N/m, is attached to thesecond block in such a way that it will be compressed when struck bythe moving block.

(a) What will be the maximum compression of the spring?

(b) What will be the final velocity of the blocks just after the collision?

Ans. (a) 0.32 m (b) −3.077 m/s ; 4.923 m/s


Chapter 10

Rotation

10.1 Section A

1. What is the angular speed of a point on Earth’s surface at latitude 40◦

N? What is the linear speed of the point?

Ans. 7.3× 10−5 rad/s, 3.5× 102 m/s

2. A centrifuge rotor is accelerated from rest to 20,000 rpm in 5.0 min.What is its average angular acceleration?

Ans. 6.98 rad/s2

3. How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm fromthe axis of rotation is to experience acceleration of 1000 g’s?

Ans. 3.57× 103 rpm

4. A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 macquires a rotational rate of 1800 rev/s from rest over a 6.0-s intervalat constant angular acceleration. Calculate the torque delivered by themotor.

Ans. 53 m.N

5. Show that 12Iω2 has units equivalent of joules.

51

CHAPTER 10. ROTATION

10.2 Section B

6. A diver makes 2.5 revolutions on the way from a 10-m-high platformto the water. Assuming zero initial vertical velocity, find the averageangular velocity during the dive.

Ans. 11 rad/s

7. A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itsangular velocity in rad/s. What are the linear speed and accelerationof a point on the edge of the grinding wheel?

Ans. 261.8 rad/s, 46 m/s, 1.2× 104 m/s2

8. The angular position of a point on a rotating wheel is given by thefunction θ = 2.0 + 4.0t2 + 2.0t3, where θ is in radians and t in seconds.At t = 0, what are

(a) the point’s angular position and

(b) its angular velocity?

(c) What is its angular velocity at t = 4.0 s?

(d) Calculate it angular acceleration at t = 2.0 s.

(e) Is its angular acceleration constant?

Ans. (a) 2.0 rad (b) 0.0 rads

(c) 128 rads

(d) 32 rads2

9. The angle through which a rotating wheel has turned in time t is givenby θ = 6.0t− 8.0t2 + 4.5t4 , where θ is in radians, and t in seconds.

(a) Determine an expression for the instantaneous angular velocity ωand for the instantaneous angular acceleration α

(b) Evaluate ω and α at t = 3.0 s.

(c) What is ω average for t ∈ (2.0 s, 3.0 s)?

Ans. (a) (6.0− 16t+ 18t3), (−16 + 54t2); (b) 444 rads

, 470 rads2

;(c) 258.5 rad

s



10. The angular acceleration of a wheel, as a function of time, is α =5.0t2 − 3.5t, where α is in rad/s2 and t in seconds. If the wheel startsfrom rest (θ = 0 and ω = 0 at t = 0), determine

(a) the angular velocity ω,

(b) and the angular position θ as a function of time.

(c) Evaluate ω and θ at t = 2.0 s.

Ans. (a) 1.67t3 − 1.75t2; (b) 0.418t4 − 0.583t3;(c) 6.4 rad/s , 2.0 rad.

11. Starting from rest, a disk rotates about its central axis with constantangular acceleration. In 5.0 s, it rotates 25 rad. During that time, whatare the magnitudes of

(a) the angular acceleration and the average velocity?

(b) What is the instantaneous angular velocity of the disk at the endof the 5.0 s?

(c) With the angular acceleration unchanged, through what addi-tional angle will the disk turn during the next 5.0 s?

Ans. (a) 2.0 rads2, 5.0 rad

s; (b) 10 rad

s; (c) 75 rad

12. A flywheel turns through 40 rev as it slows down from an angular speedof 1.5 rad/s to a stop. (Assume constant acceleration)

(a) What is its angular acceleration?

(b) Find the time for it to come to rest.

(c) *How much time is required for it to complete the first 20 of the40 revolutions?

Ans. (a) 335 s (b) (−4.5× 10−3) rads2

; (c) 98 s

13. A flywheel with a diameter of 1.20 m is rotating at an angular speedof 200 rpm.



(a) What is the angular speed of the flywheel in radians per second?

(b) What is the linear speed of a point on the rim of the flywheel?

(c) What constant angular acceleration will increase the wheel’s an-gular speed to 1000 rpm in 60.0 s?

(d) How many revolutions does the wheel make during that 60.0 s?

Ans. (a) 20.9 rad/s; (b) 12.5 m/s; (c) 800 rev/min2; (d) 600 rev

14. An object rotates about a fixed axis, and the angular position of areference line on the object is given by θ = 0.40e2t, where θ is inradians and t is in seconds. Consider a point on the object that is 4.0cm from the axis of rotation. At t = 0, what are the magnitudes of thepoint’s tangential and radial component of the acceleration?

Ans. 6.4× 10−2 m/s2, 2.56× 10−2 m/s2

15. A 0.84-m diameter solid sphere can be rotated about an axis throughits center by a torque of 10.8 m.N which accelerates it uniformly fromrest through a total of 180 revolutions in 15.0 s. What is the mass ofthe sphere?

Ans. 15 kg

p. O

ωm m m

ddd

L

Figure 10.1: Rotating Particles (Problem 16)

16. Figure 10.1 shows three 0.010 kg particles that have been glued to arod of length L = 6.0 cm and negligible mass. The assembly can rotatearound a perpendicular axis through point O at the left end. Determinethe moment of inertia of the assembly around the rotation axis.

Ans. 14md2



p.A p.Bp.O

Figure 10.2: Four Particles (Problem 17)

17. Four small spheres, each of which you can regard as a point of mass0.200 kg, are arranged in a square 0.400 m on a side and connectedby light rods as shown in Fig. 10.2. Find the moment of inertia of thesystem about an axis

(a) through the center of the square, perpendicular to its plane (anaxis through p.O in the figure).

(b) bisecting two opposite sides of the square (an axis along the lineAB in the figure).

(c) that passes through the centers of the upper left and lower rightspheres and through p.O.

Ans. (a) 0.064 kg.m2; (b) 0.032 kg.m2; (c) 0.032 kg.m2

18. Use the parallel-axis theorem to show that the rotational inertia (mo-ment of inertia) of a thin rod about an axis perpendicular to the rod atone end is I = 1

3Ml2, given that the inertia about a parallel axis going

to the center of mass is I = 112Ml2.

19. Derive the moment of inertia of a thin uniformly distributed rectangularplate about an axis through its CM and perpendicular to its plane. Thedimensions of the plate are a× b and the mass is m.

Ans.M(a2+b2)

12

20. Derive the moment of inertia of a uniform rod rotating about an axisthrough its center of mass. The mass of the rod is m and the length isL.



p. O

θ1

θ2

F2

F1

r1

r2

Figure 10.3: Two forces acting on an object (Problem 21)

21. The object in Fig. 10.3 is pivoted at p.O, and two forces act on it asshown. If r1 = 1.30 m, r2 = 2.15 m, F1 = 4.20 N, F2 = 4.90 N,θ1 = 75◦, and θ2 = 60.0◦, what is the net torque about the pivot?

Ans. 3.85 m.N , CW

F1

F2

Figure 10.4: Uniform Disk and Two Forces (Problem 22)

22. A uniform disk can rotate around its center like a merry-go-round asshown in Fig. 10.4. The disk has a radius of 2.00 cm and a mass of20.0 g and is initially at rest. Starting at time t = 0, two forces are to beapplied tangentially to the rim as indicated, so that at time t = 1.25 sthe disk has an angular velocity of 250 rad/s counterclockwise. Force~F1 has a magnitude of 0.100 N. What is the magnitude of ~F2?

Ans. 0.14 N

23. A thin rod of length 0.75 m and mass 0.42 kg is suspended freely fromone end. It is pulled to one side and then allowed to swing like apendulum, passing through its lowest position with an angular speedof 4.0 rad/s. neglecting friction and air resistance, find (a) the rod’skinetic energy at its lowest position, (b) and how far above that positionthe center of mass rises.

Ans. (a) 6.3 J (b) 0.15 m



24. A uniform meter stick is held vertically with one end on the floor andis then allowed to fall. Find the speed of the other end just before ithits the floor, assuming that the end on the floor does not slip.

Ans. 5.42 m/s

m1

m2

M, R0

h

Figure 10.5: Pulley and Two Masses (Problems 25 and 26)

25. Two masses, m1 = 35.0 kg and m2 = 38.0 kg, are connected by a ropethat hangs over a frictionless pulley, as in Fig. 10.5. The pulley is auniform solid cylinder of radius 0.30 m and mass 4.8 kg. Initially m1 ison the ground and m2 rests 2.5 m above the ground. If the system isreleased, use conservation of energy to determine the speed of m2 justbefore it strikes the ground.

Ans. 1.4 m/s

26. Two blocks, of mass m1 = 400 g and m2 = 600 g, are connectedby a massless cord that is wrapped around a uniform disk of massM = 500 g and radius R = 12.0 cm (see Fig. 10.5). The disk canbe rotated without friction about a fixed horizontal axis through itscenter; the cord cannots slip on the disk. The system is released fromrest. Find the magnitude of the acceleration of the blocks and thetension in the cord on the left and right of the pulley.

Ans. 1.57 m/s2

27. A uniform cylinder rolls down a plane inclined at an angle θ with thehorizontal. Show that if the cylinder rolls without slipping, the accel-eration is a = 2

3g sin(θ).



28. A yo-yo consists of a uniform disk with a string wound around the rim.The upper end of the string is held fixed. The yo-yo unwindes as itdrops. What is its downward acceleration?

Ans. 23g

29. A yo-yo has a rotational inertia of 950 g.cm2 and a mass of 120 g. itsaxle radius is 3.2 mm, and it string is 120 cm long. The yo-yo rollsfrom rest down to the end of the string. Determine the acceleration ofits center of mass and how long it takes to reach the end of the string.

Ans. 0.125 m/s2, 4.38 s

30. A force ~F = (16.0j − 4.0k) N is applied on a particle at location~r = (4.0i+ 8.0j + 6.0k) m. Determine the torque about the origin.

Ans. (−128, 16, 64) m.N

31. Show that the kinetic energy K of a particle of mass m, moving in acircular path is K = l2

2I, where l is the angular momentum, and I is

the moment of inertia of the particle about the center of the circle.

Pivot

250 m/s 160 m/s

Figure 10.6: Bullet Collides with a Uniform Stick (Problem 32)

32. A uniform stick 1.0 m long with a total mass of 300 g is pivoted at itscenter. A 3.0 g bullet is shot through midway between the pivot andone end, as show in Fig. 10.6. The bullet approaches at 250 m/s andleaves at 160 m/s. With what angular seed is the stick spinning afterthe collision?

Ans. 2.7 rad/s, CW



33. A diver can reduce her moment of inertia by a factor of about 3.5 whenchanging from the straight position to the tuck position. If she makes2.0 rotations in 1.5 s when in the tuck position, what is her angularspeed (rev/s) when in the straight position?

Ans. 0.38 rev/s

34. A 4.8-m-diameter merry-go-round is rotating freely with an angularvlocity of 0.80 rad/s. Its total moment of inertia is 1950 kg.m2. Fourpeople standing on the ground, each of 65 kg mass, suddenly step ontothe edge of the merry-go-round. (a) What is the angular velocity of themerry-go-round now? (b) What if the people were on it initially andthen jumped off in a radial direction (relative to the merry-go-round)?

Ans. (a) 0.45 rad/s (b) 0.80 rad/s

35. Consider the second hand of a clock to be a slender rod rotating witha constant angular speed, with mass 6.00 g and length 15.0 cm. Deter-mine its angular momentum.

Ans. 4.71× 10−6 kg.m2/ s

10.3 Section C

36. Galileo measured the acceleration of gravity by rolling a sphere doewnan inclined plane. Suppose that, starting from rest, a sphere takest = 1.6 s to roll distance of l = 3.00 m down a θ = 20◦ inclined plane.What value of g can you deduce from this?

Ans. 9.6 m/s2

37. A rod with a linear mass density λ = (0.5 kg/m) e−0.2x and a lengthL = 2.0 m, is rotating about an axis through its end. Calculate (a) themass of the rod, and (b) its rotational inertia.

Ans. (a) 0.824 kg (b) 1.0 kg.m2



A

B

R2

R1

Figure 10.7: Rotating Ball (Problem 38)

38. A ball of mass M and radius R1 on the end of a thin massless rod isrotated in a horizontal circle of radius R2 about an axis of rotation aB,as shown in Fig. 10.7

(a) Considering the mass of the ball to be concentrated at its centerof mass, calculate its moment of inertia about AB.

(b) Using the parallel axis theorem and considering the fiite radius ofthe ball, calculate the moment of inertia of the ball about AB.

(c) For R1 = 10 cm and R2 = 1.0 m, what is the percent differencebetween the values in (a) and (b)?

Ans. (a) MR22; (b) 2

5MR1

2 +MR22; (c) 0.4%

39. Use the perpendicular axis theorem to determine a formula for themoment of inertia of a thin, square plate with mass M and side sabout an axis

(a) through its center and along a diagonal of the plate

(b) through the center and parallel to a side

Ans.√2

24Ms2


Chapter 11

Statics and Equilibrium

11.1 Section A

1. A 2.0-m-long steel wire in a musical instrument has a radius of 0.03 mm.How much is the change in its length when a tension of 90 N is applied?

Ans. 0.3 m

11.2 Section B

1.0 m 2.0 m

Figure 11.1: Diving Board (Problem 2)

2. A diving board 3.00 m long is supported at a point 1.00 m from theend, and a diver weighing 500 N stands at the free end, as shown inFig. 11.1. The diving board is of uniform cross section and weighs280 N. Find the force at the support point and the force at the endthat is held down.

Ans. 1920 N, 1140 N

61

CHAPTER 11. STATICS

3. Two people carry a heavy electric motor by placing it on a light board2.00 m long. One person lifts at one end with a force of 400 N, and theother lifts the opposite end with a force of 600 N. What is the weight ofthe motor, and where along the board is its center of gravity located?

Ans. 1000 N, 1.20 m from the 400 N end

θ

Figure 11.2: A beam with a crate (Problem 4)

4. Find the tension T in each cable and the magnitude and direction ofthe force exerted on the strut by the pivot in Fig. 11.2. The weight ofthe strut and the weight of the suspendeded crate are both equal to w,and θ = 30◦.

Ans. t = 3w2 tan θ

= 2.6w, Fpivot = 3.28w at φ = 37.6◦

α β

Figure 11.3: A beam. (Problem 5)

5. Find the tension T in each cable and the magnitude and direction ofthe force exerted on the strut by the pivot in Fig. 11.3. The weight ofthe strut and the weight of the suspended crate are both equal to w,α = 30◦, and β = 45◦.

Ans. t = 4.1w, Fpivot = 5.38w at φ = 48.8◦


CHAPTER 11. STATICS

α

β


6. A uniform, 250-kg beam is supported using a cable connected to theceiling, as shown in Fig. 11.4. The lower end of the beam rests on thefloor. If α = 40◦ and β = 160◦, determine the tension in the cable, andthe minimum coefficient of static friction between the beam and thefloor required for the beam to remain in this position.

Ans. 2.74× 103 N, 19

α

β


7. A uniform, 222-N beam is attached to a wall and a cable, as shown inFig. 11.5. If the angles α and β are both equal to 30◦, determine thetension in the wire, and the horizontal and the vertical components ofthe hinge on the beam.

Ans. 192.3 N, 96.15 N, 55.46 N

8. A student takes a 100-g uniform meter stick and hangs 200 grams onthe 10-cm mark and 400 grams on the 80-cm mark. Where should thestudent clamp the meter stick so that it is in balance?

Ans. 55.7 cm


CHAPTER 11. STATICS

θ

2000 N

L/4

3L/4

p.P


9. A uniform, 400-N beam is supported by a rope and a hinge as shownin Fig. 11.6. If θ = 50◦, find the tension in the tie rope and the forceon the beam by the pin at p.P.

Ans. 2460 N, 3.44 kN at θ = 44◦

10. A circular steel wire 2.00 m long must stetch no more than 0.25 cm whena tensile force of 400 N is applied to each end of the wire (That is 400 Napplied on one end and 400 N applied on the other end simultaneously).What minimum diameter is required for the wire? (E = 2×1011 N/m2)

Ans. 2.02 mm

T

F

Figure 11.7: Three pulleys. (Problem 11)

11. The force ~F in Fig. 11.7 keeps the 6.40-kg block and the pulleys inequilibrium. The pulleys have negligible mass and friction. Calculatethe tension T in the upper cable.

Ans. 71.68 N


CHAPTER 11. STATICS

α

β

2.50 m 0.50 m

Figure 11.8: Hanging Monkey (Problem 12)

12. A 3.00-m-long, 240-N uniform rod at the zoo is held in a horizontalposition by two ropes at its ends, as shown in Fig. 11.8. The left ropemakes an angle of α = 150◦ with the rod and the right rope makes anangle β with the horizontal. A 90-N monkey hangs motionless 0.50 mfrom the right end of the rod. Calculate the tension in the two ropesand the angle β.

Ans. Tl = 270 N, Tr = 304 N, and β = 40◦

CM

3.2 m

1.65 mx

Mg

FL

FT

FD

Figure 11.9: An Airplane (Problem 13)

13. The forces acting on a 77,000-kg aircraft flying at constant velocity areshown in Fig. 11.9. The engine thrust, FT = 5.0× 105 N, acts on a line1.6 m below the CM. Determine the drag force FD and the distance xabove the CM that it acts.


CHAPTER 11. STATICS

Ans. 3.18 m

14. A 40-N uniform plank leans against a frictionless wall, as shown inFig. 11.10. Determine the force exerted by the wall on the board.

Ans. 10 N

3 m

6 m

Figure 11.10: A Uniform Plank (Problem 14)

15. A 400-N ball is suspended on a string AB, as shown in Fig. 11.11. Ifthe ball rests against a frictionless wall and the string make an angle30◦ with the wall, determine the tension in the string and the forceexerted by the wall.

Ans.

θ

A

B

Figure 11.11: A Uniform Plank (Problem 15)


CHAPTER 11. STATICS

11.3 Section C

16. A door 1.00 m wide and 2.00 m high weighs 280 N and is supportedby two hinges, one 0.50 m from the top and the other 0.50 m fromthe bottom. Each hinge supports half the total weight of the door.Assuming that the door’s center of gravity is at its centr, find thehorizontal components of the force exerted on the door by each hinge.

Ans. 140 N


Chapter 12

Gravitation

12.1 Section A

1. Suppose the Earth were compressed to half its diameter. (a) Whatwould be the acceleration due to gravity at its surface? (b) How fastwould it be rotating about its axis?

Ans. (a) 39.2 ms2

; (b) 96 hr

2. What would be the gravitational acceleration on Jupiter?

Ans. 24.8 m/s2

3. Determine the escape velocity for the Moon.

Ans. 2.37 km/s

12.2 Section B

4. The asteroid belt between Mars and Jupiter consists of many fragments,once hypothesized to have been a planet. If the center of mass of theasteroid belt is about 3 times farther from the Sun than the Earth is,how long would it have taken this hypothetical planet to orbit the Sun?

Ans. 5.196 yr

68

CHAPTER 12. GRAVITATION

5. Sputnik I, the first artificial satellite to circle the planet (Oct. 1957)had a mean orbital radius of 6950 km. What was its period?

Ans. 96.2 min

6. Comet Halley moves in an elongated elliptical orbit around the sun. Atperihelion (closest distance), the comet is 0.586 AU from the sun; ataphelion (the farthest distance), the comet is 35.1 AU. Find the periodof the comet.

Ans. 76 yr

7. Suppose that a planet were discovered between the Sun and Mercury,with a circular orbital of radius equal to 2

3of the average orbit radius

of Mercury (R = 0.387 AU). What would be the orbital period of sucha planet?

Ans. 0.13 yr

8. A satellite circles planet Zeron every 98 min. The mass of this planetis known to be 5.0× 1024 kg. What is the radius of the orbit?

Ans. 6.6× 106 m

9. For a satellite to be in a circular orbit 789 km above the surface ofthe earth, (a) what orbital speed must it be given, and (b) what is theperiod of the orbit (in hours)

Ans. (a) 7.46× 103 m/s; (b) 1.68 h

10. An asteroid with mass m = 5 × 109 kg approaching the Earth. At adistance 5.0× 106 km from the Earth it speed is 600 m/s. What wouldits speed be at 6.38 × 103 km from the Earth? (Neglect any frictionwith the Earth’s atmosphere.)

Ans. 11.17 km/s



11. A meteor (m = 575 kg) has a speed of 90.0 m/s when 800 km abovethe Earth. It is falling vertically (ignore air resistance) and strikes abed of sand in which it is brought to rest in 3.25 m.

(a) What is its speed just before striking the sand?

(b) How much work does the sand do to stop the meteor ?

(c) What is the average force exerted by the sand on the meteor?

(d) How much thermal energy is produced?

Ans. (a) 3.7× 103 m/s; (b) −3.98× 109 J; (c) 1.23× 109 N;(d) 3.98× 109 J;

12. The Earth’s distance from the sun varies from 1.471×108 km to 1.521×108 km during the year. Determine the difference in

(a) the potential energy,

(b) the Earth’s kinetic energy,

(c) the total energy between these extreme points.

Ans. (a) −1.7× 1032 J; (b) 0.85× 1032 J; (c) −0.85× 1032 J;

13. On July 4, 2005, the NASA spacecraft Deep Impact fired a projectileonto the surface of Comet Tempel 1. This comet is about 9.0 km across.Observations of surface debris released by the impact showed that dustwitha speed as low as 1.0 m/s was able to escape the comet. Assuminga spherical shape, what is the mass of this comet?

Ans. 3.75× 1013 kg

14. A rocket is accelerated to speed v = 2√gRe near Earth’s surface, and

it then coasts upward. How much will be its velocity when it is farfrom Earth?

Ans.√

2gRe



15. A projectile is launched from the surface of a planet with a mass Mand radius R. What minimum launch speed is required if the projectileis to rise to a height of 2R above the surface of the planet? Disregardany dissipative effects of the atmosphere.

Ans.√

4GM3R

16. Venus has a diameter of 12.1× 103 km and a mean density of 5.2× 103

kg/m3. How far would an apple fall in one second at its surface?

Ans. 4.4 m

17. [Note1 ]A pendulum with a period of oscillations of 3.0 seconds on Earthis taken to the Moon. ( M Moon = 0.0122M Earth, R Moon = 0.272R Earth)What would be its period there?

Ans. 7.4 s

12.3 Section C

18. Certain neutron stars are believed to be rotating at about 1 rev/s. Ifsuch a star has a radius of 20 km, what must be its minimum mass sothat material on its surface remains in place during the rapid rotation?

Ans. 4.74× 1024 kg

19. If Earth were to lose its orbital velocity and began falling “down” di-rectly towards the Sun, how long would it take for it to reach the Sun?

Ans. 0.35 yr

1Skip if Oscillations have not be covered in class yet


Chapter 13

Fluids

13.1 Group A

water

oilhoil

hwater

Figure 13.1: U-shaped Tube (Problem 1)

1. Water and then oil (which don’t mix) are poured into a U-shaped tube,open at both ends. They come to equilibrium as shown in Fig. 13.1.What is the density of the oil?

Ans. h waterρ water

h oil

2. One arm of a U-shaped tube (open at both ends) contains water andthe other alcohol (ρ = 0.79×103 kg/m3). If the two fluids meet exactlyat the bottom of the U, and the alcohol is at a height of 18.0 cm, atwhat height will the water be?

Ans. 14.22 cm

72

CHAPTER 13. FLUIDS

3. A hydraulic press for compacting powdered samples has a large cylinderwhich is 10.0 cm in diameter, and a small cylinder with a diameter of2.0 cm. A force of 300 N is applied to the small cylinder. The sample,which is placed on the large cylinder, has an area of 4.0 cm2. What isthe pressure on the sample?

Ans. 1.875× 107 N/m2

4. The deepest point known in any of the earth’s oceans is in the MarianasTrench, 10.92 km deep. Assuming water to be incompressible, what isthe pressure at this depth? (Use density of seawater 1027 kg/m3).

Ans. 1.1× 108 Pa

13.2 Section B

5. A barrel contains a 0.120-m-layer of oil floating on water that is 0.250 mdeep. The density of the oil is 600 kg/m3. What is the gauge pressure(a) at the oil-water surface?, and (b) at the bottom of the barrel?

Ans. (a) 706 Pa (b) 3160 Pa

58°

110 m

5 m

Figure 13.2: House Pressure (Problem 6)

6. A house at the bottom of a hill is fed by a full tank of water 5.0 m deepand connected to the house by a pipe that is 110 m long at an angleof 58◦ from the horizontal, as shown in Fig. 13.2. Determine the watergauge pressure at the house.

Ans. 9.63× 105 Pa


CHAPTER 13. FLUIDS

7. A slab of ice (ρ = 0.91 gcm3 ) floats on a freshwater lake. What minimum

volume must the slab have for a 45.0-kg woman to be able to stand onit without getting her feet wet?

Ans. 0.54 m3

Wood

Copper

Figure 13.3: Floating Copper Block (Problem 9)

8. If the maximum pressure difference a person’s lungs can withstandwithout interruption to the breathing process is 85 mm− Hg, deter-mine how deep a person can stay underwater breathing through a longsnorkel.

Ans. 1.12 m

9. A copper weight is placed on top of a 0.50 kg block of wood (ρ =0.5× 103 kg/m3) floating in water, as shown in Fig. 13.3. What is themass of the copper if the top of the wood block is exactly at the watersurface?

Ans. 0.5 kg

10. An iron anchor of density 7870 kg/m3 appears 200 N lighter in waterthan in air. What is its mass?

Ans. 161 kg

11. A 9.28-kg rock appears to have have a mass of 6.18 kg when submergedin water. What is the density of the rock?

Ans. 3.0× 103 kg/m3


CHAPTER 13. FLUIDS

12. What is the likely identity of a metal if a sample has a mass of 63.5 gwhen measured in air and an apparent mass of 55.4 g when submergedin water?

Ans. iron or steel

13. A 3.40-kg aluminum ball has an apparent mass of 2.10 kg when sub-merged in a particular liquid. Calculate the density of the fluid.

Ans. 1.03× 103 kg/m3

Fb

mloadg

mHeg

Figure 13.4: Balloon (Problem 14)

14. A spherically shaped balloon has a radius of 7.35 m, and is filled withhelium, as shown in Fig. 13.4. How large a cargo can it lift, assumingthat the skin and structure of the balloon have a mass of 1000 kg? Ne-glect the buoyant force on the cargo volume itself. (ρ helium = 0.1787 kg

m3 ,

ρ air = 1.29 kgm3 )

Ans. 850 kg

15. A 15-cm radius air duct is used to replenish the air of a room 9.2 m ×5.0 m × 4.5 m every 16 min. How fast does air flow in the duct?

Ans. 3.1 m/s


CHAPTER 13. FLUIDS

16. A shower head has 20 circular openings, eachwith radus 1.0 mm. Theshower head is connected to a pipe with radius 0.80 cm. If the speed ofwater in the pipe is 3.0 m/s, what is its speed as it exits the shower-headopenings?

Ans. 9.6 m/s

17. A garden hose with an internal diameter 1.9 cm is connected to astationar lawn sprinkler that consists merely of an enclosure with 24holes, each 0.13 cm in diameter. If the water in the hose has a speedof 0.91 m/s, at what speed does it leave the sprinkler holes?

Ans. 8.1 m/s

18. A sealed tank containing seawater to a height of 11.0 m also containsair above the water at a gauge pressure of 3.00 atm. Water flows outfrom the bottom through a small hole. Calculate the speed of the waterwith which it flows out. (Density of seawater: ρ = 1027 kg

m3 )

Ans. 28.7 m/s

19. How fast does water flow from a hole at the bottom of a very wide,4.6-m-deep storage tank filled with water? Ignore viscosity.

Ans. 9.5 m/s

20. A tank of large area is filled with water to a depth D = 0.30 m. A holeof cross-sectional area A = 65 cm2 in the bottom of the tank allowswater to drain out.

(a) What is the rate at which water flows out, in cubic meters persecond?

(b) At what distance below the bottom of the tank is the cross-sectional area of the stream equal to one half of the area of thehole?

Ans. (a) 0.0158 m3/ s, (b) 0.90 m


CHAPTER 13. FLUIDS

21. What is the lift (in Newton) due to Bernoulli’s principle on a wing ofarea 86 m2 if the air passes over the top and bottom surfaces at speedsof 340 m/s and 290 m/s, respectively.

Ans. 1.76× 106 N

22. If the speed of flow past the lower surface of an airplane wing is 110m/s, what speed of flow over the upper surface will give a pressuredifference of 900 Pa betwen upper and lower surfaces? Take density ofair to be 1.3× 10−3 g/cm3.

Ans. 116 m/s

23. Air streams horizontally past a small airplane’s wings such that thespeed is 70.0 m/s over the top surface and 60.0 m/s past the bottomsurface. If the plane has mass 1340 kg and a total wing area of 16.2m2, what is the net vertical force (inluding the effects of gravity) onthe airplane? The density of air is 1.2 kg

m3

Ans. 496 N downward

13.3 Section C

hH

Figure 13.5: Water Tank (Problem 24)

24. Water stands at a depth H in a large, open tank whose side walls arevertical. A hole is made in one of the walls at a depth h below thewater surface, as shown in Fig. 13.5. At what distance R from the footof the wall does the emergin stream strike the floor?

Ans. 2√h(H − h)


CHAPTER 13. FLUIDS

25. A cylindrical bucket of liquid (density ρ) is rotated about its symmetryaxis, which is vertical. If the angular velocity is ω, show that thepressure at a distance r from the rotational axis is P = Po + 1

2ρ2r2,

where Po is the pressure at r = 0.


Chapter 14

Oscillations

14.1 Section A

1. The displacement from equillibrium of an oscillating particle is givenby y(t) = (5.0) cos (3t+ π

2), where the distances are given in meters

and time in seconds. Determine:

(a) the amplitude of oscillations,

(b) the angular frequency,

(c) the frequency of oscillations,

(d) the period of oscillations,

(e) the velocity of the particle as a function of time,

(f) the maximum acceleration.

Ans. (a) 5.0 m (b) 3 rad/s (c) 0.48 Hz (d) 2.1 s(e) (−15) sin (3t+ π

2) (f) 45.0 m/s2

2. A pendulum makes 42 vibrations in 50 s. What are its period andfrequency?

Ans. 1.19 s, 0.84 Hz

3. Find the mechanical energy of a block-spring system having a springconstant of 1.3 N/cm and an oscillation amplitude of 2.4 cm.

Ans. 37 mJ

79

CHAPTER 14. OSCILLATIONS

14.2 Section B

4. A 2-kg particle oscillates on a string with a coefficient of stiffness equalto 50 N/m. At time equal t = 1.0 s, the particle is at maximumdisplacement from equilibrium equal to 5.0 cm. Determine the functionthat describes the position of the particle as a function of time and themaximum acceleration of the particle.

Ans. 1.25 m/s2

5. A small fly of mass 0.60 g is caught in a spider’s web. The web vibratespredoinantly with a frquency of 10 Hz.

(a) What is the value of the effective spring constant k for the web?

(b) At what frequency would you expect the web to virbrate if aninsect of mass 0.40 g were trapped?

Ans. (a) 2.37 N/m (b) 12.25 Hz

6. A spring vibrates with a frequency of 3.0 Hz when a weight of 0.50 kgis hung from it. What will be the frequency if 0.35 kg hangs from it?

Ans. 3.6 Hz

7. The length of a simple pendulum is 0.68 m and it is released at anangle of 12◦ to the vertical.

(a) With what frequency does it vibrate?

(b) What is the pendulum bob’s speed when it passes through thelowest point of the swing?

Ans. (a) 0.6 Hz (b) 0.54 m/s

8. If a particle oscillates in a simple harmonic motion with an amplitudeA, at what discplacement from equilibrium is its speed equal to halfthe maximum value?

Ans. 0.866A



9. A 450-g particle oscillates from a vertically hanging light spring onceevery 0.55 s. Assuming it started by being compressed 10 cm from thequilibrium position (where y = 0), and released,

(a) Write down an equation giving its position y as a function of t?

(b) What will be its maximum speed and maximum acceleration?

Ans. (a) y = (0.1) cos (11.4t); (b) 1.1 m/s, 13 m/s2

10. A block with mass m is attached to a spring with a spring constant k.Initially, the block is at maximum displacement. How long does it takethe block to go back to the equilibrium position?

Ans. π2

√mk

11. A 0.0125-kg bullet strikes a 0.300-kg block attached to a fixed horizontalspring whose spring constant is 2.25×103 N/m and sets it into vibrationwith an amplitude of 12.4 cm. What was the speed of the bullet if thetwo objects move together after impact?

Ans. 263 m/s

14.3 Section C


12. A mass m is connected to two springs with equal spring constants k asshown on Fig. 14.1. The mass is pulled a distance ∆a to the right andbegins to oscillate. What is the period of oscillations?

Ans. 2π√

m2k



k1

k2

m


13. A block of mass m is supported by two prallel vertical springs withspring constant k1 an k2 as shown in Fig. 14.2. What will be thefrequency of vibrations?

Ans. 12π

√k1+k2m

14. [Note1 ]If we took a simple pendulum with length 1.0 m to Mars, whatwould be its period of oscillations?

Ans. 3.3 s

ρwater

H ρ

Wood


15. A rectangular block of wood with density ρ and height H, floats ina container of water with a density ρ water. Show that, if friction isignored, (a) when the block is pushed gently down into the water, itwill then vibrate with simple harmonic motion, (b) and determine theperiod of oscillations.

Ans. 2π√

ρ watergρH

16. Derive a formula for the maximum speed v◦ of a simple pendulum bobin terms of g, the length L, and the maximum angle of swing θ◦.

1Skip if Gravitation is not covered in class yet



Ans. θo√gL

17. A 750-g blok oscillates on the end of a spring whose force constant isk = 56.0 N/m. The mass moves in a fluid which offers a resistive forceF = −bv, where b = 0.162 N.s/m

(a) What is the period of motion?

(b) What is the fractional decrease in amplitude per cycle?

(c) Write the displacement as a function of time if at t = 0 x = 0,and at t = 1.00 s x = 0.120 m.

Ans. (a) 0.733 s (b) 45.3%

(c) (0.4684 m) e−(1.08 1s)t cos

(8.57 rad

st− π

2

)


Chapter 17

Temperature, Heat, ThermalExpansion, Ideal Gas Laws

17.1 Section A

1. How many atoms are there in 3.4-g copper penny?

Ans. 3.2× 1022

2. What is the molar mass of methanol, CH3OH? What is the number ofmolecules in 1.0 kg of methanol?

Ans. 32; 1.88× 1025

17.2 Section B

3. At what temperature will the Fahrenheit and Centigrade scales yieldthe same numerical value?

Ans. −40◦

4. What mass of steam at 100◦C must be added to 1.00 kg of ice at 0◦Cto yield liquid water at 20◦C?

Ans. 0.16 kg

84

CHAPTER 17. TEMPERATURE AND HEAT

5. A 1.00-liter container made of glass is full to the brim with ethyl alcoholat −110◦C (near its freezing point). How many cubic centimeters ofalcohol overflow the container if the system is heated to +75◦C (nearboiling point)? (Ignore the expansion of the glass; coeff. of volumeexpansion for ethyl alcohol: β = 0.00109 1

◦C)

Ans. 202 mL

6. A 600-W stove burner is turned on full to heat up an iron frying panto 400◦C. How long will it take for the pan to reach that temperatureif its mass is 5.0 kg and it is initially at 25◦C?

Ans. 24 min

7. A small 200-Watt electric immersion heater is used to heat 100 g of wa-ter for a cup of instant coffee. Calculate the time required to bring thewater from 23◦C to 100◦C ignoring any thermal energy that transfersout of the cup.

Ans. 1.6× 102 s

8. Warm tea at 50◦C is poured over ice to make iced tea. The specificheat of tea is the same as that of water. If 500 mL of tea are to cooledto 5.0◦C, how many grams of ice at 0.0◦C are neeeded?

Ans. 2.7× 102 g

9. You place 1.0 kg of ice (at 0◦C) in a pot and heat it until the ice meltsand the water boils off, making steam. How much heat must you supplyto achieve this?

Ans. 719.2 Cal

10. Ballistic Pendulum An iron bullet (m = 10 g, c = 0.46 kJ/kg.C) isfired into a wooden block (M = 2.0 kg, c = 2 kJ/kg.C) with a speed ofv = 150 m/s. The bullet embeds itself in the block and they both swingup at height H above their initial position. Determine (a) how highthey swing, H, and (b) by how much do their temperature changes inthe process



Ans. 2.8 cm; 0.28◦C

11. The height of the Eiffel Tower is 321 m. What increase of temperaturewill lead to an increase of height by 10 cm? (Consider it made of steel;α = 12× 10−6 1

◦C)

Ans. 26◦C

12. In an alcohol-in-glass thermometer, the alcohol column has length 11.82cm at 0.0◦C and length 22.85 cm at 100◦C. What is the temperatureif the column has length (a) 16.70 cm, and (b) 20.50 cm

Ans. 44.2◦C; 78.7◦C

13. A metal rod is 40.125 cm long at 20◦C and 40.148 cm long at 45.0◦C.Calculate the average coefficient of linear expansion of the rod for thistemperature range.

Ans. 2.3× 10−5 1◦C

14. The pendulum shaft of a clock is made of brass (α = 2.0 × 10−5 1◦C

).What is the fractional change in length of the shaft when it is cooledfrom 20.0◦C to 5.00◦C?

Ans. −0.03%

15. The volume of air in the fully expanded human lungs is 5.0 L. Assumethat the air is 76% nitrogen and 24% oxygen by mass, at a temperature37◦C and a pressure of 1.0 atm. (a) How many molecules are in thelungs? (b) How many molecules of oxygen and how many of nitrogen?

Ans. (a) 1.18× 1023; (b) 9× 1022 N, 2.8× 1022 O

16. If 3.00 m3 of a gas initially at STP is placed under pressure of 3.20atm, The temperature of the gas rises to 38.0◦C. What is the volumeof the gas?

Ans. 1.07 m3



17. A storage tank contains 21.6 kg of molecular nitrogen ( N2) at an ab-solute pressure of 3.65 atm. What will the pressure be if the nitrogenis replaced by an equal mass of carbon dioxide, CO2 at the same tem-perature.?

Ans. 2.32 atm

17.3 Section C

18. A pendulum clock keeps good time when its temperature is 15◦C. If itsshaft is made of brass (α = 2×10−5 1

◦C, how much time (in seconds per

day) will the clock lose when its temperature is 35◦C?

Ans. 17.3 s

19. You can warm the surfaces of your hands by rubbing them one againstthe other. If the coefficient of friction between your hands is 0.60 and ifyou press your hands togeher with a force of 60 N while rubbing themback and forth at an average speed of 0.50 m/s, at what rate (in joulesper second) do you generate heat on the surfaces of your hands?

Ans. 18 W

20. An iron cube (β = 12 × 10−6 1◦C

) floats in a bowl of liquid mercury(β = 1.8 × 10−4 1

◦C) at 0◦C. If the temperature is raised to 25◦C, by

what percent will the fraction of volume submerged change?

Ans. 0.42%


Chapter 18

Kinetic Theory of Gases

18.1 Section A

1. What is the thermal kinetic energy in 1.0 kg oxygen gas at 20◦C? Whatfraction of that energy is translational?

Ans. 1.9× 105 J; 0.6

18.2 Section B

2. A gas cylinder contains 30 liters of diatomic nitrogen gas at 273 K anda pressure of 140 atm. If the temperature is increased to 300 K, howmuch will the internal energy of the gas increase?

Ans. 1.0× 105 J

3. What is the average kinetic energy of an oxygen molecule in air atSTP?

Ans. 5.65× 10−21 J

4. Using laser beams, physicists can cool a small amount of a gas of sodiumatoms to extremely low temperature. Determine the rms speed of asodium atom when such laser cooling results in a temperatuer of T =2.0× 10−4 K.

88

CHAPTER 18. KINETIC THEORY

Ans. 0.47 m/s

5. Calculate the rms speed of Helium atoms near the surface of the Sunat a temperature of 6000K.

Ans. 6.1× 103 m/s

18.3 Section C

6. A group of 25 particles have the following speeds: two have a speed10 m/s, seven have 15 m/s, four have 20 m/s, three have 25 m/s, sixhave 30 m/s, one has 35 m/s, and two have 40 m/s. Determine, (a) theaverage speed, (b) the rms speed, and (c) the most probably speed.

Ans. (a) 23 m/s; (b) 25 m/s; (c) 15 m/s

7. At 273 K and 1.00× 10−2 atm, the density of a gas is 1.24× 10−5 gcm3 .

(a) Find the molar mass of the gas and identify it. (b) Find the v rms

for the gas molecules.

Ans. (a) N2 (b) 493 m/s

8. Show that for a mixture of two gases at the same temperature, theratio o their rms speeds is equal to the inverse ratio of the square rootsof their moleculal masses.

9. The two isotopes of uranium, 235U and 238U, can be separated by a gasdiffusion process by combining them with fluorine (µ = 19) to makethe gaseous compound UF6. Calculate the ratio of the rms speeds ofthese molecules for the two isotopes.

Ans. 1.00429


Chapter 19

First Law of Thermodynamics

19.1 Section A

1. An ideal monatomic gas absorbs 2600 J of heat and performs 900 J ofwork. By how much has its internal energy changed?

Ans. 1700 J

2. Five moles of an ideal monatomic gas with an initial temperature of127◦C expand and, in the process, absorb 1200 J of heat and do 2100 Jof work. What is the final temperature of the gas?

Ans. 113◦C

3. During an isothermal compression of an ideal gas, 335 J of het must beremoved from the gas to maintain constant temperature. How muchwork is done by the gas during the process?

Ans.

4. The process shown in Fig. 19.1 involves 0.0175 mole of an ideal diatomicgas.

(a) What was the lowest temperature the gas reached in this process?

(b) Find the work in the process AB.

(c) Find the work in the process BC.

Ans. 278 K, 0 J, 162 J

90

CHAPTER 19. FIRST LAW

2.0 4.0 6.0 V (L)

P (Atm)

0.20

0.40

0.60

A

B

C

Figure 19.1: PV Diagram (Problem 4)

19.2 Section B

5. Five moles of gas have their tempearature increased from −10◦C to+20◦C. How much is the heat transferred into the gas if the gas is

(a) He at constant pressure of 1.5 atm.

(b) Ar at a fixed volume of 8.2 m3.

(c) CO2 at a constant pressure of 20 kPa.

Ans. (a) 3.12 kJ (b) 1.87 kJ (c) 4.37 kJ

6. An ideal gas initially at 4.00 atm and 350 K is premitted to expandadiabatically to 1.50 times the initial volume. Find the final pressureand temperature if the gas is (a) monatomic (b) diatomic

Ans. (a) 2.04 atm, 267 K (b) 2.27 atm, 298 K

7. During an adiabatic expansion the temperature of 0.450 moles of Ardrops from 50.0◦C to 10.0◦C. The argon may be treated as an idealgas. (a) How much is the work by the gas? (b) What is the change inthe internal energy of the gas?

Ans. (a) 224.5 J (b) -224.5 J



1.0 2.0 3.0 Volume (m3)

Pressure (kPa) 7.5

5.0

2.5 a

b

c


8. A sample of an ideal gas is taken through the cyclic process abca shownin Fig. 19.2. (T a = 200 K)

(a) How many moles of gas are there in the sample?

(b) What are the temperatures T b, and T c?

(c) What is the change in the internal energy of the sample?

(d) How much work is done by gas?

(e) How much heat has flown?

Ans. (a) 1.5; (b) 1800 K, 600 K, (c) 0 J; (d) +5 kJ; (e) +5 kJ

9. 4.65-mol sample of an ideal diatomic gas expands adiabatically from avolume of 0.120 m3 to 0.750 m3. Initially the pressure was 1.00 atm.Assuming no molecular vibrations, determine (a) the initial and finaltemperatures, (b) the change in the internal energy, (c) the heat lostby the gas, and (d) the work done by the gas.

Ans. (a) 314 K, 151 K; (b) −15.8 kJ; (c) 0 (d) +15.8 kJ

10. A three-step process is shown on the diagram in Fig. 19.3, in whichmonatomic ideal gas is initially in the state A with P = 2.5 atm,V = 6 L, and T = 300 K. It is allowed to expand isothermally toP = 1.5 atm and V = 10 L (state B). It is then compressed slowlykeeping the pressure constant to its initial volume V1 = 6 L (state C),and eventually the pressure is increased to its initial value P1 (state A),while keeping the volume constant.



V1 V2

P1

P2

A

B

C


(a) How much is the heat during each proces: AB, BC, and CA?

(b) How much is the work during each process: AB, BC, and CA?

Ans. (a) 776 J, −1.52× 103 J, 913 J; (b) 776 J, -608 J, 0;

11. Propane gas (C3H8) behaves like an ideal gas with γ = 1.127. Deter-mine the molar heat capacity at constant volume and the molar heatcapacity at constant pressure.

Ans. 65.5 Jmol.K

, 73.8 Jmol.K


Chapter 20

Entropy and Second Law ofThermodynamics

20.1 Section A

1. A 10.0-kg box having an initial speed of 3.0 m/s slides along a roughtable and comes to rest. Estimate the total change in entropy of theuniverse. Assume all objects are at room temperature (293 K)

Ans. 0.15 J/K

20.2 Section B

2. A nuclear power plant operates at 75 pecent of its maximum theoret-ical efficiency between temperature of 660◦C and 360◦C. If the plantgenerates electric energy at the rate of 1.1 GW, how much exhaust heatis discharged per hour?

Ans. 1.25× 1013 J/h

3. A three-step process is shown on the diagram in Fig. 20.1, in whichmonatomic ideal gas is initially in the state A with P = 2.5 atm,V = 6 L, and T = 300 K. It is allowed to expand isothermally toP = 1.5 atm and V = 10 L (state B). It is then compressed slowlykeeping the pressure constant to its initial volume V1 = 6 L (state C),

94

CHAPTER 20. SECOND LAW

V1 V2

P1

P2

A

B

C


and eventually the pressure is increased to its initial value P1 (state A),while keeping the volume constant.

(a) What is the entropy change during processes: AB, BC, and CA?

Ans. +2.58733 J/K; -2.58674 J/K; +2.58674 J/K

(b) What is the maximum theoretical efficiency for such a cycle?

(c) What is the actual efficiency of the cycle?

Ans. 40%; 7.8%

4. A diatomic gas is used as an engine in a Carnot cycle ( a → b → c →d → a). If n = 0.5 mol, Va = 6.0 L, Vb = 15.0 L, TH = 470◦C, andTL = 290◦C, calculate:

(a) the pressure and the volume in the states a and b;

(b) the volume in the states c and d.

(c) What is the work done during the process ab?

(d) What is the heat in the process cd?

(e) Calculate the net work done for the whole cycle.

(f) Use the definition of the efficiency as e = |W ||QL|

to calculate theefficiency of the cycle.

Ans. (a) 5.15× 105 Pa, 2.06× 105 Pa; (b) 30.0 L, 12.0 L;(c) 2.83× 103 J; (d) −2.14× 103 J; (e) 0.69× 103 J; and (f) 24%


CHAPTER 20. SECOND LAW

5. Calculate the change in entropy of 1.00 kg of water when it is heatedfrom 0◦C to 100◦C.

Ans. 0.312 kcal/K

6. A 3.8-kg piece of aluminum at 30◦C is placed in 1.0 kg of water in aStyrofoam container at room temperature (20◦C). Estimate the netchange in entropy of the system.

Ans. +2.74× 10−4 kcalK

20.3 Section C

7. Show that the change in entropy in a Carnot cycle is zero


Appendix A

Heats of Selected Substances

Substance Specific Heat Fusion VaporizationkJ/kg kcal/kg kJ/kg kJ/kg

Aluminum 0.90 0.215Brass 0.380 0.092

Copper 0.39 0.092Gold 0.126 0.0301Iron 0.46 0.108Lead 0.13 0.031Silver 0.233 00558

Mercury 0.140 0.033Water 4.186 1 334 2258

Ice (−10◦ C) 2.05 0.45C2H5OH 2.4 0.58Granite 0.79 0.19Glass 0.84 0.20

97

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