(b) The magnitude of the displacement from start to finish. (c) The

displacement from start to finish.

(b) The magnitude of the displacement from start to finish. (c) The

displacement from start to finish.

(b) The magnitude of the displacement from start to finish. (c) The

displacement from start to finish.

(b) The magnitude of the displacement from start to finish. (c) The

displacement from start to finish.

average velocity over a period of one year?

5.00 m from the center of rotation. (a) Calculate the average speed of the

blade tip in the helicopter’s frame of reference. (b) What is its average

velocity over one revolution?

rate of about 3 cm/y. At this rate how long will it take them to drift 500 km

farther apart than they are at present?

an average velocity of about 6 cm/y northwest relative to land east of the

fault. Los Angeles is west of the fault and may thus someday be at the

same latitude as San Francisco, which is east of the fault. How far in the

future will this occur if the displacement to be made is 590 km northwest,

assuming the motion remains constant?

Zephyr set the world’s nonstop long-distance speed record for trains. Its

run from Denver to Chicago took 13 hours, 4 minutes, 58 seconds, and

was witnessed by more than a million people along the route. The total

distance traveled was 1633.8 km. What was its average speed in km/h

and m/s?

of the Moon is increasing in radius at a rate of approximately 4 cm/year.

Assuming this to be a constant rate, how many years will pass before the

odometer reading of her car increased by 12.0 km. The trip took 18.0

min. (a) What was her average speed? (b) If the straight-line distance

east, what was her average velocity? (c) If she returned home by the

same path 7 h 30 min after she left, what were her average speed and

velocity for the entire trip?

in a nerve cell depends (inversely) on the diameter of the axon (nerve

fiber). If the nerve cell connecting the spinal cord to your feet is 1.1 m

long, and the nerve impulse speed is 18 m/s, how long does it take for

the nerve signal to travel this distance?

characterized by a kind of echo in which the earthbound person’s voice

was so loud in the astronaut’s space helmet that it was picked up by the

astronaut’s microphone and transmitted back to Earth. It is reasonable to

assume that the echo time equals the time necessary for the radio wave

to travel from the Earth to the Moon and back (that is, neglecting any time

delays in the electronic equipment). Calculate the distance from Earth to

the Moon given that the echo time was 2.56 s and that radio waves travel

2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He

breaks the tackle and runs straight forward another 21.0 m in 5.20 s.

Calculate his average velocity (a) for each of the three intervals and (b)

for the entire motion.

atomic nucleus much as planets orbit the Sun. In this model you can view

hydrogen, the simplest atom, as having a single electron in a circular

number of revolutions per second it makes about the nucleus. (b) What is

the electron’s average velocity?

What is its acceleration?

Dr. John Paul Stapp was U.S. Air Force officer who studied the effects of

extreme deceleration on the human body. On December 10, 1954, Stapp

rode a rocket sled, accelerating from rest to a top speed of 282 m/s (1015

km/h) in 5.00 s, and was brought jarringly back to rest in only 1.40 s!

Calculate his (a) acceleration and (b) deceleration. Express each in

gravity.

(b) If she then brakes to a stop in 0.800 s, what is her deceleration?

suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are

position vs. time for this period.

from the time the ball first touches the mitt until it stops, what was the

initial velocity of the ball?

What is its muzzle velocity (that is, its final velocity)?

How long does it take to reach its top speed of 80.0 km/h, starting from

CHAPTER 2 | KINEMATICS 81

How long does it take to come to a stop from its top speed? (c) In

emergencies the train can decelerate more rapidly, coming to rest from

knowns in this problem. (c) How far does the car travel in those 12.0 s?

To solve this part, first identify the unknown, and then discuss how you

chose the appropriate equation to solve for it. After choosing the

equation, show your steps in solving for the unknown, check your units,

and discuss whether the answer is reasonable. (d) What is the car’s final

velocity? Solve for this unknown in the same manner as in part (c),

showing all steps explicitly.

(b) What is her final velocity? (c) Evaluate the result. Does it make

sense?

Blood is accelerated from rest to 30.0 cm/s in a distance of 1.80 cm by

the left ventricle of the heart. (a) Make a sketch of the situation. (b) List

the knowns in this problem. (c) How long does the acceleration take? To

solve this part, first identify the unknown, and then discuss how you

chose the appropriate equation to solve for it. After choosing the

equation, show your steps in solving for the unknown, checking your

units. (d) Is the answer reasonable when compared with the time for a

heartbeat?

8.00 m/s to 40.0 m/s in the same direction. If this shot takes

h) in only 3.90 s. (a) What is its average acceleration? (b) How far does it

travel in that time?

decelerations. (a) What is the final velocity of a freight train that

velocity of 4.00 m/s? (b) If the train can slow down at a rate of

(c) How far will it travel in each case?

over a distance of 0.250 m. (a) How long did the acceleration last? (b)

Calculate the acceleration.

top of the water. (a) If the swan must reach a velocity of 6.00 m/s to take

far will it travel before becoming airborne? (b) How long does this take?

A woodpecker’s brain is specially protected from large decelerations by

tendon-like attachments inside the skull. While pecking on a tree, the

woodpecker’s head comes to a stop from an initial velocity of 0.600 m/s

. (b) Calculate the stopping time. (c) The

tendons cradling the brain stretch, making its stopping distance 4.50 mm

(greater than the head and, hence, less deceleration of the brain). What

running at a velocity of 7.50 m/s and comes to a full stop after

compressing the padding and his body 0.350 m. (a) What is his

deceleration? (b) How long does the collision last?

jumped from their flaming airplanes with no parachute to escape certain

death. Some fell about 20,000 feet (6000 m), and some of them survived,

with few life-threatening injuries. For these lucky pilots, the tree branches

and snow drifts on the ground allowed their deceleration to be relatively

small. If we assume that a pilot’s speed upon impact was 123 mph (54

m/s), then what was his deceleration? Assume that the trees and snow

stopped him over a distance of 3.0 m.

ignore air resistance in this case (only for the sake of this problem),

determine a squirrel’s velocity just before hitting the ground, assuming it

fell from a height of 3.0 m. (b) If the squirrel stops in a distance of 2.0 cm

through bending its limbs, compare its deceleration with that of the

airman in the previous problem.

through. The station is 210 m long. (a) How long is the nose of the train in

the station? (b) How fast is it going when the nose leaves the station? (c)

If the train is 130 m long, when does the end of the train leave the

station? (d) What is the velocity of the end of the train as it leaves?

the final velocity of this dragster starting from rest and accelerating at the

rate found in (a) for 402 m (a quarter mile) without using any information

on time. (c) Why is the final velocity greater than that used to find the

acceleration is valid for a dragster. If not, discuss whether the

acceleration would be greater at the beginning or end of the run and what

effect that would have on the final velocity.

racer has an initial velocity of 11.5 m/s and accelerates at the rate of

continues at this velocity to the finish line. If he was 300 m from the finish

line when he started to accelerate, how much time did he save? (c) One

other racer was 5.00 m ahead when the winner started to accelerate, but

he was unable to accelerate, and traveled at 11.8 m/s until the finish line.

How far ahead of him (in meters and in seconds) did the winner finish?

motorcycle, on the Bonneville Salt Flats in Utah, of 183.58 mi/h. The oneway

course was 5.00 mi long. Acceleration rates are often described by

the time it takes to reach 60.0 mi/h from rest. If this time was 4.00 s, and

Burt accelerated at this rate until he reached his maximum speed, how

long did it take Burt to complete the course?

Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt “coasted”

across the finish line with a time of 9.69 s. If we assume that Bolt

accelerated for 3.00 s to reach his maximum speed, and maintained that

speed for the rest of the race, calculate his maximum speed and his

acceleration. (b) During the same Olympics, Bolt also set the world

record in the 200-m dash with a time of 19.30 s. Using the same

assumptions as for the 100-m dash, what was his maximum speed for

this race?

1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial

1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down

with an initial velocity of 14.0 m/s from the Verrazano Narrows Bridge in

New York City. The roadway of this bridge is 70.0 m above the water.

At what velocity must a basketball player leave the ground to rise 1.25 m

above the floor in an attempt to get the ball?

One of the rescuers throws a life preserver straight down to the victim

with an initial velocity of 1.40 m/s and observes that it takes 1.8 s to

82 CHAPTER 2 | KINEMATICS

reach the water. (a) List the knowns in this problem. (b) How high above

the water was the preserver released? Note that the downdraft of the

helicopter reduces the effects of air resistance on the falling life

preserver, so that an acceleration equal to that of gravity is reasonable.

velocity of 13.0 m/s. (a) List the knowns in this problem. (b) How high

does his body rise above the water? To solve this part, first note that the

final velocity is now a known and identify its value. Then identify the

unknown, and discuss how you chose the appropriate equation to solve

for it. After choosing the equation, show your steps in solving for the

unknown, checking units, and discuss whether the answer is reasonable.

(c) How long is the dolphin in the air? Neglect any effects due to his size

or orientation.

into a pool. She starts with a velocity of 4.00 m/s, and her takeoff point is

1.80 m above the pool. (a) How long are her feet in the air? (b) What is

her highest point above the board? (c) What is her velocity when her feet

hit the water?

ground when it is thrown straight up from the cliff with an initial velocity of

8.00 m/s. (b) How long would it take to reach the ground if it is thrown

straight down with the same speed?

with an initial velocity of 11.0 m/s. How long does he have to get out of

the way if the shot was released at a height of 2.20 m, and he is 1.80 m

tall?

passes a tree branch on the way up at a height of 7.00 m. How much

additional time will pass before the ball passes the tree branch on the

way back down?

vertical speed when it leaves the ground. (b) How long is it in the air?

Australia, a hiker hears a rock break loose from a height of 105 m. He

can’t see the rock right away but then does, 1.50 s later. (a) How far

above the hiker is the rock when he can see it? (b) How much time does

he have to move before the rock hits his head?

Determine the distance traveled during the first second. (b) Determine

the final velocity at which the object hits the ground. (c) Determine the

distance traveled during the last second of motion before hitting the

ground.

California. Suppose a boulder breaks loose from the top of this cliff. (a)

How fast will it be going when it strikes the ground? (b) Assuming a

reaction time of 0.300 s, how long will a tourist at the bottom have to get

out of the way after hearing the sound of the rock breaking loose

(neglecting the height of the tourist, which would become negligible

anyway if hit)? The speed of sound is 335 m/s on this day.

off the ground on its path up and takes 1.30 s to go past the window.

What was the ball’s initial velocity?

equipment, you measure the time for the sound of a splash to return. (a)

Neglecting the time required for sound to travel up the well, calculate the

distance to the water if the sound returns in 2.0000 s. (b) Now calculate

the distance taking into account the time for sound to travel up the well.

The speed of sound is 332.00 m/s in this well.

rebounds to a height of 1.45 m. (a) Calculate its velocity just before it

strikes the floor. (b) Calculate its velocity just after it leaves the floor on its

way back up. (c) Calculate its acceleration during contact with the floor if

compress during its collision with the floor, assuming the floor is

absolutely rigid?

ground and rising at 10.0 m/s upward. For the coin, find (a) the maximum

height reached, (b) its position and velocity 4.00 s after being released,

and (c) the time before it hits the ground.

and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it

strikes the floor. (b) Calculate its velocity just after it leaves the floor on its

way back up. (c) Calculate its acceleration during contact with the floor if

compress during its collision with the floor, assuming the floor is

absolutely rigid?

Note: There is always uncertainty in numbers taken from graphs. If your

answers differ from expected values, examine them to see if they are

within data extraction uncertainties estimated by you.

CHAPTER 2 | KINEMATICS 83

acceleration and velocity given in the examples for this figure.

average acceleration between 0 and 4 s? (d) What is his time for the

race?

the corresponding velocity and acceleration graphs.

84 CHAPTER 2 | KINEMATICS

1. Find the following for path A in Figure 3.54: (a) the total distance

Figure 3.54 The various lines represent paths taken by different people walking in a

2. Find the following for path B in Figure 3.54: (a) the total distance

3. Find the north and east components of the displacement for the hikers

shown in Figure 3.52.

4. Suppose you walk 18.0 m straight west and then 25.0 m straight north.

you represent the two legs of the walk as vector displacements A and

B , as in Figure 3.55, then this problem asks you to find their sum

R = A + B .)

Figure 3.55 The two displacements A and B add to give a total displacement R

having magnitude R and direction θ .

5. Suppose you first walk 12.0 m in a direction 20º west of north and

then 20.0 m in a direction 40.0º south of west. How far are you from

the walk as vector displacements A and B , as in Figure 3.56, then this

problem finds their sum R = A + B .)

6. Repeat the problem above, but reverse the order of the two legs of the

B , which is 20.0 m in a direction exactly 40º south of west, and then

leg A , which is 12.0 m in a direction exactly 20º west of north. (This

problem shows that A + B = B + A .)

7. (a) Repeat the problem two problems prior, but for the second leg you

walk 20.0 m in a direction 40.0º north of east (which is equivalent to

subtracting B from A —that is, to finding R′ = A − B ). (b) Repeat

direction 40.0º south of west and then 12.0 m in a direction 20.0º east

of south (which is equivalent to subtracting A from B —that is, to

finding R′′ = B - A = - R′ ). Show that this is the case.

8. Show that the order of addition of three vectors does not affect their

sum. Show this property by choosing any three vectors A , B , and C ,

all having different lengths and directions. Find the sum A + B + C

the same. (There are five other orders in which A , B , and C can be

9. Show that the sum of the vectors discussed in Example 3.2 gives the

result shown in Figure 3.24.

10. Find the magnitudes of velocities vA and vB in Figure 3.57

Figure 3.57 The two velocities vA and vB add to give a total vtot .

11. Find the components of vtot along the x- and y-axes in Figure 3.57.

12. Find the components of vtot along a set of perpendicular axes

rotated 30º counterclockwise relative to those in Figure 3.57.

13. Find the following for path C in Figure 3.58: (a) the total distance

Figure 3.58 The various lines represent paths taken by different people walking in a

14. Find the following for path D in Figure 3.58: (a) the total distance

15. Find the north and east components of the displacement from San

Francisco to Sacramento shown in Figure 3.59.

16. Solve the following problem using analytical techniques: Suppose you

two legs of the walk as vector displacements A and B , as in Figure

3.60, then this problem asks you to find their sum R = A + B .)

Figure 3.60 The two displacements A and B add to give a total displacement R

having magnitude R and direction θ .

17. Repeat Exercise 3.16 using analytical techniques, but reverse the

same result—that is, B + A = A + B .) Discuss how taking another

18. You drive 7.50 km in a straight line in a direction 15º east of north.

19. Do Exercise 3.16 again using analytical techniques and change the

second leg of the walk to 25.0 m straight south. (This is equivalent to

subtracting B from A —that is, finding R′ = A – B ) (b) Repeat

again, but now you first walk 25.0 m north and then 18.0 m east. (This

is equivalent to subtract A from B —that is, to find A = B + C . Is

20. A new landowner has a triangular piece of flat land she wishes to

displacement vectors A from B in Figure 3.61. She then correctly

calculates the length and orientation of the third side C . What is her

21. You fly 32.0 km in a straight line in still air in the direction 35.0º

direction 45.0º south of west and then in a direction 45.0º west of

set of axes—one rotated 45º .

22. A farmer wants to fence off his four-sided plot of flat land. He

measures the first three sides, shown as A, B, and C in Figure 3.62,

D . What is his result?

23. In an attempt to escape his island, Gilligan builds a raft and sets to

the following straight lines: 2.50 km 45.0º north of west; then

4.70 km 60.0º south of east; then 1.30 km 25.0º south of west;

then 5.10 km straight east; then 1.70 km 5.00º east of north; then

7.20 km 55.0º south of west; and finally 2.80 km 10.0º north of

24. Suppose a pilot flies 40.0 km in a direction 60º north of east and

then flies 30.0 km in a direction 15º north of east as shown in Figure

3.63. Find her total distance R from the starting point and the direction

θ of the straight-line path to the final position. Discuss qualitatively how

25. A projectile is launched at ground level with an initial speed of 50.0

m/s at an angle of 30.0º above the horizontal. It strikes a target above

the ground 3.00 seconds later. What are the x and y distances from

26. A ball is kicked with an initial velocity of 16 m/s in the horizontal

27. A ball is thrown horizontally from the top of a 60.0-m building and

28. (a) A daredevil is attempting to jump his motorcycle over a line of

buses parked end to end by driving up a 32º ramp at a speed of

40.0 m/s (144 km/h) . How many buses can he clear if the top of the

29. An archer shoots an arrow at a 75.0 m distant target; the bull’s-eye of

30. A rugby player passes the ball 7.00 m across the field, where it is

31. Verify the ranges for the projectiles in Figure 3.41(a) for θ = 45º

32. Verify the ranges shown for the projectiles in Figure 3.41(b) for an

33. The cannon on a battleship can fire a shell a maximum distance of

curved. Assume that the radius of the Earth is 6.37×103 km . How

34. An arrow is shot from a height of 1.5 m toward a cliff of height H . It

is shot with a velocity of 30 m/s at an angle of 60º above the horizontal.

35. In the standing broad jump, one squats and then pushes off with the

this position is 1.25 times the acceleration due to gravity, g . How far can

36. The world long jump record is 8.95 m (Mike Powell, USA, 1991).

37. Serving at a speed of 170 km/h, a tennis player hits the ball at a

height of 2.5 m and an angle θ below the horizontal. The service line is

11.9 m from the net, which is 0.91 m high. What is the angle θ such that

38. A football quarterback is moving straight backward at a speed of 200

the ball is thrown at an angle of 25º relative to the ground and is caught

39. Gun sights are adjusted to aim high to compensate for the effect of

40. An eagle is flying horizontally at a speed of 3.00 m/s when the fish in

41. An owl is carrying a mouse to the chicks in its nest. Its position at that

nest. The owl is flying east at 3.50 m/s at an angle 30.0º below the

42. Suppose a soccer player kicks the ball from a distance 30 m toward

2.4 m above the ground, given the initial direction to be 40º above the

43. Can a goalkeeper at her/ his goal kick a soccer ball into the

44. The free throw line in basketball is 4.57 m (15 ft) from the basket,

45. In 2007, Michael Carter (U.S.) set a world record in the shot put with

at a height of 2.10 m and threw it at an angle of 38.0º above the

ground is achieved at 45º when air resistance is neglected, the actual

angle to achieve maximum range is smaller; thus, 38º will give a longer

range than 45º in the shot put.)

46. A basketball player is running at 5.00 m/s directly toward the basket

47. A football player punts the ball at a 45.0º angle. Without an effect

48. Prove that the trajectory of a projectile is parabolic, having the form

y = ax + bx2 . To obtain this expression, solve the equation x = v0x t

for t and substitute it into the expression for y = v0yt – (1 / 2)gt2

(These equations describe the x and y positions of a projectile that

y = ax + bx2 where a and b are constants.

49. Derive R = v0 2 sin 2θ0 g for the range of a projectile on level ground

by finding the time t at which y becomes zero and substituting this

value of t into the expression for x − x0 , noting that R = x − x0

50. Unreasonable Results (a) Find the maximum range of a super

51. Construct Your Own Problem Consider a ball tossed over a fence.

52. Bryan Allen pedaled a human-powered aircraft across the English

He flew for 169 min at an average velocity of 3.53 m/s in a direction 45º

53. A seagull flies at a velocity of 9.00 m/s straight into the wind. (a) If it

54. Near the end of a marathon race, the first two runners are separated

55. Verify that the coin dropped by the airline passenger in the Example

3.8 travels 144 m horizontally while falling 1.50 m in the frame of

56. A football quarterback is moving straight backward at a speed of 2.00

ball is thrown at an angle of 25.0º relative to the ground and is caught at

relative to the quarterback ?

57. A ship sets sail from Rotterdam, The Netherlands, heading due north

direction 40.0º north of east. What is the velocity of the ship relative to

58. A jet airplane flying from Darwin, Australia, has an air speed of 260

m/s in a direction 5.0º south of west. It is in the jet stream, which is

blowing at 35.0 m/s in a direction 15º south of east. What is the velocity

59. (a) In what direction would the ship in Exercise 3.57 have to travel in

speed relative to the water remains 7.00 m/s ? (b) What would its speed

60. (a) Another airplane is flying in a jet stream that is blowing at 45.0 m/s

in a direction 20º south of east (as in Exercise 3.58). Its direction of

motion relative to the Earth is 45.0º south of west, while its direction of

travel relative to the air is 5.00º south of west. What is the airplane’s

61. A sandal is dropped from the top of a 15.0-m-high mast on a ship

62. The velocity of the wind relative to the water is crucial to sailboats.

in a direction 30.0º east of north relative to the Earth. It encounters a

wind that has a velocity of 4.50 m/s in a direction of 50.0º south of west

63. The great astronomer Edwin Hubble discovered that all distant

we are at the center of an expanding universe. Figure 3.64 illustrates this

Figure 3.64 Five galaxies on a straight line, showing their distances and velocities

64. (a) Use the distance and velocity data in Figure 3.64 to find the rate

65. An athlete crosses a 25-m-wide river by swimming perpendicular to

66. A ship sailing in the Gulf Stream is heading 25.0º west of north at a

4.80 m/s 5.00º west of north. What is the velocity of the Gulf Stream?

67. An ice hockey player is moving at 8.00 m/s when he hits the puck

The line between the center of the goal and the player makes a 90.0º

angle relative to his path as shown in Figure 3.65. What angle must the

Figure 3.65 An ice hockey player moving across the rink must shoot backward to give

68. Unreasonable Results Suppose you wish to shoot supplies straight

69. Unreasonable Results A commercial airplane has an air speed of

280 m/s due east and flies with a strong tailwind. It travels 3000 km in a

direction 5º south of east in 1.50 h. (a) What was the velocity of the

70. Construct Your Own Problem Consider an airplane headed for a

What is the net external force on him?

prin, gen:  1a.  A bicycle and its rider experience a net force of 250 N, which accelerates them at 2.3 m/s^2.  What is the total mass of the bicycle and its rider?

m, and then maintains that velocity for the remainder of the 100-m dash,

what will be his time for the race?

external force on it is 60.0 N. Calculate its acceleration.

measuring their masses is needed to monitor their mass gains or losses

to adjust diets. One way to do this is to exert a known force on an

astronaut and measure the acceleration produced. Suppose a net

external force of 50.0 N is exerted and the astronaut’s acceleration is

force on the astronaut, the vehicle in which they orbit experiences an

equal and opposite force. Discuss how this would affect the

measurement of the astronaut’s acceleration. Propose a method in which

recoil of the vehicle is avoided.

newtons) is the person exerting on the mower? Suppose the mower is

go before stopping?

Assume that the rockets are off. The mass of the system is 2100 kg.

burning, what is its acceleration? Assume that the mass of the system is

2100 kg, and the force of friction opposing the motion is known to be 650

N. (b) Why is the acceleration not one-fourth of what it is with all rockets

burning?

from a speed of 1000 km/h? (Such deceleration caused one test subject

to black out and have temporary blindness.)

directions, on a third child in a wagon. The first child exerts a force of

75.0 N, the second a force of 90.0 N, friction is 12.0 N, and the mass of

the third child plus wagon is 23.0 kg. (a) What is the system of interest if

the acceleration of the child in the wagon is to be calculated? (b) Draw a

free-body diagram, including all forces acting on the system. (c) Calculate

the acceleration. (d) What would the acceleration be if friction were 15.0

N?

while traveling at 90.0 km/h. At that speed the forces resisting motion,

including friction and air resistance, total 400 N. (Air resistance is

analogous to air friction. It always opposes the motion of an object.) What

force does the motorcycle exert backward on the ground to produce its

acceleration if the mass of the motorcycle with rider is 245 kg?

prin, gen: 10a.  What is the force required to accelerate a 10 gram pellet from rest to a velocity of 100 m/s in a rifle barrel 0.6 cm long?

gen: 10b.  A fish is being pulled upward. The breaking strength of the line holding the fish is 22 N. An acceleration of  2.5 m/s^2 breaks the line. What can we say about the mass of the fish?

horizontal component of the force the seat exerts against his body.

Compare this with his weight by using a ratio. (b) Calculate the direction

and magnitude of the total force the seat exerts against his body.

exerted by the seat and restraining belts.

250 N. How much do they weigh on Earth? What is the mass on the

Moon? On Earth?

off from the Moon is 10,000 kg. The thrust of its engines is 30,000 N. (a)

Calculate its acceleration in a vertical takeoff from the Moon. (b) Could it

lift off from Earth? If not, why not? If it could, calculate its acceleration.

force is exerted on the ship by the artillery shell?

opposing player who is exerting a force of 800 N on him. The mass of the

losing player plus equipment is 90.0 kg, and he is accelerating at

player’s feet and the grass? (b) What force does the winning player exert

on the ground to move forward if his mass plus equipment is 110 kg? (c)

Draw a sketch of the situation showing the system of interest used to

solve each part. For this situation, draw a free-body diagram and write

the net force equation.

first team’s members has an average mass of 68 kg and exerts an

average force of 1350 N horizontally. Each of the second team’s

members has an average mass of 73 kg and exerts an average force of

1365 N horizontally. (a) What is the acceleration of the two teams? (b)

What is the tension in the section of rope between the teams?

independent of the velocity of the gymnast—she can be moving either up

or down, or be stationary.

tension in a horizontal strand of spider web if the same spider sits

156 CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION

this with the tension in the vertical strand (find their ratio).

the rope if he climbs at a constant speed? (b) What is the tension in the

medium at its center and perpendicular to its length (such as on the

mass of the child and basket if a scale reading of 55 N is observed? (b)

the scale has a mass of 0.500 kg? (d) Draw a sketch of the situation

indicating the system of interest used to solve each part. The masses of

the cords are negligible.

What is the rocket’s acceleration? Explicitly show how you follow the

steps in the Problem-Solving Strategy for Newton’s laws of motion.

road to accelerate the car in the forward direction. If the force of friction

including air resistance is 250 N and the acceleration of the car is

show how you follow the steps in the Problem-Solving Strategy for

Newton’s laws of motion. For this situation, draw a free-body diagram

and write the net force equation.

produce an upward acceleration 4.00 times the acceleration due to

gravity. Explicitly show how you follow the steps in the Problem-Solving

Strategy for Newton’s laws of motion.

25b:  A person of mass 66 kg crouches then jumps to a height of .8 meters. From the crouched position to the point where the person leaves the ground the distance is 20 cm. What average force is exerted over this 20-cm distance?

decelerates by pushing straight down on the mat. Calculate the force she

must exert if her deceleration is 7.00 times the acceleration due to

gravity. Explicitly show how you follow the steps in the Problem-Solving

Strategy for Newton’s laws of motion.

engine exert backward on the track to accelerate the train at a rate of

engines exert identical forces? This is not a large frictional force for such

a massive system. Rolling friction for trains is small, and consequently

trains are very energy-efficient transportation systems. (b) What is the

force in the coupling between the 37th and 38th cars (this is the force

each exerts on the other), assuming all cars have the same mass and

that friction is evenly distributed among all of the cars and engines?

loading area by a tractor. (a) An 1800-kg tractor exerts a force of

forces resisting motion that total 2400 N. If the acceleration is

exerted by the tractor on the airplane, assuming 2200 N of the friction is

experienced by the airplane. (c) Draw two sketches showing the systems

of interest used to solve each part, including the free-body diagrams for

each.

motion of the car, boat, and trailer, if the car exerts a 1900-N force on the

boat plus trailer is 700 kg. (b) What is the force in the hitch between the

car and the trailer if 80% of the resisting forces are experienced by the

boat and trailer?

graphically or by using trigonometry. (b) Show graphically that the same

part (b) or a similar picture.

of the 49.00-kg sled and child system. Note that the direction of the

frictional force is unspecified; it will be in the opposite direction of the sum

CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION 157

sled.

would you have to exert perpendicular to the center of the rope to

produce a force of 12,000 N on the car if the angle is 2.00°? In this part,

explicitly show how you follow the steps in the Problem-Solving Strategy

for Newton’s laws of motion. (b) Real ropes stretch under such forces.

What force would be exerted on the car if the angle increases to 7.00°

and you still apply the force found in part (a) to its center?

wire is 25.0 N? Note that the force applied to the tooth is smaller than the

tension in the wire, but this is necessitated by practical considerations of

how force can be applied in the mouth. Explicitly show how you follow

steps in the Problem-Solving Strategy for Newton’s laws of motion.

figure are the tensions applied by the wire to the protruding tooth. The total force

motionless from a rope. Superhero’s mass is 90.0 kg, while Trusty

Sidekick’s is 55.0 kg, and the mass of the rope is negligible. (a) Draw a

free-body diagram of the situation showing all forces acting on

Superhero, Trusty Sidekick, and the rope. (b) Find the tension in the rope

above Superhero. (c) Find the tension in the rope between Superhero

and Trusty Sidekick. Indicate on your free-body diagram the system of

interest used to solve each part.

figure out what to do next. Will the tension be the same everywhere in the rope?

mass of 28.0 kg, and the force of friction is 60.0 N. (a) Draw a free-body

diagram for the system of interest. (b) What force must the nurse exert to

move at a constant velocity?

cable during the time the elevator starts from rest and accelerates its load

upward to some cruising velocity. Taking the elevator and its load to be

the system of interest, draw a free-body diagram. Then calculate the

tension in the cable. Among the things to consider are the mass of the

elevator and its load, the final velocity, and the time taken to reach that

velocity.

toboggan with four children on it up a snow-covered slope. Construct a

problem in which you calculate the acceleration of the toboggan and its

load. Include a free-body diagram of the appropriate system of interest as

the basis for your analysis. Show vector forces and their components and

explain the choice of coordinates. Among the things to be considered are

the forces exerted by those pushing, the angle of the slope, and the

masses of the toboggan and children.

the result? (c) Which premise is unreasonable, and why is it

unreasonable?

unreasonable about the result? (This result has been unintentionally

achieved by several real rockets.) (c) Which premise is unreasonable, or

which premises are inconsistent? (You may find it useful to compare this

problem to the rocket problem earlier in this section.)

158 CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION

the ground. A breeze blowing on the flea parallel to the ground exerts a

neglect the gravitational force.

and lateral heads of the gastrocnemius muscle.) Find the magnitude and

direction of the total force on the Achilles tendon. What type of movement

could be caused by this force?

is momentarily motionless. Include a free-body diagram in your solution.

more vertical rope supports a greater part of her weight (a vertical force).

7.50 m/s in 2.30 s to join another dolphin in play. What average force was

exerted to slow him if he was moving horizontally? (The gravitational

force is balanced by the buoyant force of the water.)

exerts an average force of 650 N backward on the ground for 0.800 s. (a)

What is his final speed? (b) How far does he travel?

initial acceleration if it takes off vertically. (b) How long does it take to

reach a velocity of 120 km/h straight up, assuming constant mass and

thrust? (c) In reality, the mass of a rocket decreases significantly as its

fuel is consumed. Describe qualitatively how this affects the acceleration

and time for this motion.

To do this, he lowers his body 0.300 m and then accelerates through this

distance by forcefully straightening his legs. This player leaves the floor

with a vertical velocity sufficient to carry him 0.900 m above the floor. (a)

Calculate his velocity when he leaves the floor. (b) Calculate his

acceleration while he is straightening his legs. He goes from zero to the

velocity found in part (a) in a distance of 0.300 m. (c) Calculate the force

he exerts on the floor to do this, given that his mass is 110 kg.

from a mortar and reaches a height of 110 m. (a) Neglecting air

resistance (a poor assumption, but we will make it for this example),

calculate the shell’s velocity when it leaves the mortar. (b) The mortar

itself is a tube 0.450 m long. Calculate the average acceleration of the

shell in the tube as it goes from zero to the velocity found in (a). (c) What

is the average force on the shell in the mortar? Express your answer in

newtons and as a ratio to the weight of the shell.

of 1700 kg. (a) The elevator accelerates upward from rest at a rate of

elevator. (b) The elevator continues upward at constant velocity for 8.50

s. What is the tension in the cable during this time? (c) The elevator

CHAPTER 4 | DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION 159

the cable during deceleration? (d) How high has the elevator moved

above its original starting point, and what is its final velocity?

s? (b) What is unreasonable about the result? (c) Which premise is

unreasonable, or which premises are inconsistent?

in an elevator that accelerates from rest to 30.0 m/s in 2.00 s. (a)

Calculate the scale reading in newtons and compare it with his weight.

(The scale exerts an upward force on him equal to its reading.) (b) What

is unreasonable about the result? (c) Which premise is unreasonable, or

which premises are inconsistent?

strong nuclear force? (b) What is the strength of the weak nuclear force

relative to the electromagnetic force? Since the weak nuclear force acts

at only very short distances, such as inside nuclei, where the strong and

electromagnetic forces also act, it might seem surprising that we have

any knowledge of it at all. We have such knowledge because the weak

nuclear force is

1. A physics major is cooking breakfast when he notices that the frictional

2. (a) When rebuilding her car’s engine, a physics major must exert 300

3. (a) What is the maximum frictional force in the knee joint of a person

4. Suppose you have a 120-kg wooden crate resting on a wood floor. (a)

5. (a) If half of the weight of a small 1.00×103 kg utility truck is

6. A team of eight dogs pulls a sled with waxed wood runners on wet

7. Consider the 65.0-kg ice skater being pushed by two others shown in

Figure 5.21. (a) Find the direction and magnitude of Ftot , the total force

exerted on her by the others, given that the magnitudes F1 and F2 are

direction of Ftot ? (c) What is her acceleration assuming she is already

moving in the direction of Ftot ? (Remember that friction always acts in

the direction opposite that of motion or attempted motion between

surfaces in contact.)

8. Show that the acceleration of any object down a frictionless incline that

makes an angle θ with the horizontal is a = g sin θ . (Note that this

9. Show that the acceleration of any object down an incline where friction

behaves simply (that is, where fk = μkN ) is

a = g( sin θ − μkcos θ). Note that the acceleration is independent of

friction becomes negligibly small (μk = 0).

10. Calculate the deceleration of a snow boarder going up a 5.0º , slope

result of Exercise 5.1 may be useful, but be careful to consider the fact

steps in Problem-Solving Strategies.

11. (a) Calculate the acceleration of a skier heading down a 10.0º slope,

result of Exercise 5.1 to be useful. Explicitly show how you follow the

steps in the Problem-Solving Strategies.

12. If an object is to rest on an incline without slipping, then friction must

will not slide down is θ = tan–1 μs . You may use the result of the

previous problem. Assume that a = 0 and that static friction has

13. Calculate the maximum deceleration of a car that is heading down a

6º slope (one that makes an angle of 6º with the horizontal) under the

On wet concrete. (c) On ice, assuming that μs = 0.100 , the same as

14. Calculate the maximum acceleration of a car that is heading up a 4º

slope (one that makes an angle of 4º with the horizontal) under the

(c) On ice, assuming that μs = 0.100 , the same as for shoes on ice.

15. Repeat Exercise 5.2 for a car with four-wheel drive.

16. A freight train consists of two 8.00×105-kg engines and 45 cars

with average masses of 5.50×105 kg . (a) What force must each

5.00×10−2 m / s2 if the force of friction is 7.50×105 N , assuming

17. Consider the 52.0-kg mountain climber in Figure 5.22. (a) Find the

Figure 5.22 Part of the climber’s weight is supported by her rope and part by friction

18. A contestant in a winter sporting event pushes a 45.0-kg block of ice

across a frozen lake as shown in Figure 5.23(a). (a) Calculate the

minimum force F he must exert to get the block moving. (b) What is its

19. Repeat Exercise 5.3 with the contestant pulling the block of ice with

shown in Figure 5.23(b).

Figure 5.23 Which method of sliding a block of ice requires less force—(a) pushing or

20. The terminal velocity of a person falling in air depends upon the

0.140 m2 .

21. A 60-kg and a 90-kg skydiver jump from an airplane at an altitude of

22. A 560-g squirrel with a surface area of 930 cm2 falls from a 5.0-m

23. To maintain a constant speed, the force provided by a car’s engine

Toyota Camry? (Drag area is 0.70 m2 ) (b) What is the drag force at 70

km/h and 100 km/h for a Hummer H2? (Drag area is 2.44 m2 ) Assume

24. By what factor does the drag force on a car increase as it goes from

25. Calculate the velocity a spherical rain drop would achieve falling from

across of the drop to be 4 mm, the density to be 1.00×103 kg/m3 , and

the surface area to be πr2 .

26. Using Stokes’ law, verify that the units for viscosity are kilograms per

27. Find the terminal velocity of a spherical bacterium (diameter

2.00 μm ) falling in water. You will first need to note that the drag force is

to be 1.10×103 kg/m3 .

28. Stokes’ law describes sedimentation of particles in liquids and can be

Suppose a steel ball bearing (density 7.8×103 kg/m3 , diameter

3.0 mm ) is dropped in a container of motor oil. It takes 12 s to fall a

29. During a circus act, one performer swings upside down hanging from

30. During a wrestling match, a 150 kg wrestler briefly stands on one

31. (a) The “lead” in pencils is a graphite composition with a Young’s

modulus of about 1×109 N / m2 . Calculate the change in length of the

32. TV broadcast antennas are the tallest artificial structures on Earth. In

33. (a) By how much does a 65.0-kg mountain climber stretch her

34. A 20.0-m tall hollow aluminum flagpole is equivalent in strength to a

35. As an oil well is drilled, each new section of drill pipe supports its own

36. Calculate the force a piano tuner applies to stretch a steel piano wire

37. A vertebra is subjected to a shearing force of 500 N. Find the shear

38. A disk between vertebrae in the spine is subjected to a shearing force

of 1×109 N / m2 . The disk is equivalent to a solid cylinder 0.700 cm

39. When using a pencil eraser, you exert a vertical force of 6.00 N at a

mm in diameter and is held at an angle of 20.0º to the horizontal. (a) By

40. To consider the effect of wires hung on poles, we take data from

Example 4.8, in which tensions in wires supporting a traffic light were

calculated. The left wire made an angle 30.0º below the horizontal with

41. A farmer making grape juice fills a glass bottle to the brim and caps it

a way that the volume increases by 0.2% (that is, ΔV / V0 = 2×10−3 )

per square centimeter if its bulk modulus is 1.8×109 N/m2 , assuming

42. (a) When water freezes, its volume increases by 9.05% (that is,

ΔV / V0 = 9.05×10−2 ). What force per unit area is water capable of

43. This problem returns to the tightrope walker studied in Example 4.6,

who created a tension of 3.94×103 N in a wire making an angle 5.0º

44. The pole in Figure 5.24 is at a 90.0º bend in a power line and is

line. The tension in each line is 4.00×104 N , at the angles shown. The

top of the pole at an angle of 30.0º with the vertical. (Clearly, the guy

of soup he pushes 0.600 m horizontally with a force of 5.00 N? Express

your answer in joules and kilocalories.

work done to accomplish this task.

lift it 40.0 m at constant speed, assuming friction averages 100 N. (b)

What is the work done on the lift by the gravitational force in this

process? (c) What is the total work done on the lift?

of gasoline. Only 30% of the gasoline goes into useful work by the force

keep the car moving at constant speed? (b) If the required force is

directly proportional to speed, how many gallons will be used to drive 108

km at a speed of 28.0 m/s?

pulls them as shown.

below the horizontal. (a) What is the work done on the cart by friction? (b)

What is the work done on the cart by the gravitational force? (c) What is

the work done on the cart by the shopper? (d) Find the force the shopper

exerts, using energy considerations. (e) What is the total work done on

the cart?

with that of an 80.0-kg astronaut in orbit moving at 27,500 km/h.

energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the

larger energies needed for the movement of larger animals would relate

to metabolic rates.

= 1 nautical mile/h).

non-panic stop).

(b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 2.00 m. Calculate the force exerted on the car and compare it with the force found in part (a).

with an immovable object without damage to the body of the car. The

bumper cushions the shock by absorbing the force over a distance.

Calculate the magnitude of the average force on a bumper that collapses

0.200 m while bringing a 900-kg car to rest from an initial speed of 1.1

m/s.

the force exerted by a boxing glove on an opponent’s face, if the glove

and face compress 7.50 cm during a blow in which the 7.00-kg arm and

glove are brought to rest from an initial speed of 10.0 m/s. (b) Calculate

the force exerted by an identical blow in the gory old days when no

gloves were used and the knuckles and face would compress only 2.00

cm. (c) Discuss the magnitude of the force with glove on. Does it seem

high enough to cause damage even though it is lower than the force with

no glove?

sprinter exerts backward on the track to accelerate from 2.00 to 8.00 m/s

in a distance of 25.0 m, if he encounters a headwind that exerts an

average force of 30.0 N against him.

lake has an average height of 50 m above the generators?

(b) Compare this with the energy stored in a 10-megaton fusion bomb.

CHAPTER 7 | WORK, ENERGY, AND ENERGY RESOURCES 255

which it is built) is stored in the Great Pyramid of Cheops, given that its

surrounding ground? (b) How does this energy compare with the daily

food intake of a person?

snake and raises it 2.5 m from the ground to a branch. (a) How much

work did the bird do on the snake? (b) How much work did it do to raise

its own center of mass to the branch?

descended 20.0 m was only slightly greater when it had an initial speed

of 5.00 m/s than when it started from rest. This implies that

final speed of the toy car is 0.687 m/s if its initial speed is 2.00 m/s and it

coasts up the frictionless slope, gaining 0.180 m in altitude.

compared with the gain in gravitational potential energy on even small hills.) To demonstrate this, find the final speed and the time taken for a

answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events. 

(Note:  On a 30 degree slope the skier descends half a meter for every meter she moves forward, and the component of the gravitational force in the direction down the incline is half her weight.)

0.500 m/s in 0.400 m by a large spring bumper at the end of its track.

which can be compressed 12.0 cm. To what maximum height can a child

jump on the stick using only the energy in the spring, if the child and stick

have a total mass of 40.0 kg? Explicitly show how you follow the steps in

that the coefficient of friction between her skis and the snow is 0.0800.

(Hint: Find the distance traveled up the incline assuming a straight-line

path as shown in the figure.) 

[The normal force between the incline and the skier is about 82% of her weight and the frictional force is 8% of that.  It is recommended that you show how the figure for the normal force is obtained using trigonometry, if at this point you know how to do so.]

[Principles of Physics students can assume negligible friction]

rise.

done by friction is negligible and its initial speed is 110 km/h? (b) If, in

actuality, a 750-kg car with an initial speed of 110 km/h is observed to

coast up a hill to a height 22.0 m above its starting point, how much

thermal energy was generated by friction? (c) What is the average force

broken by the energy carried by a single electron in the beam of an old-fashioned

TV tube? (These electrons were not dangerous in themselves,

but they did create dangerous x rays. Later model tube TVs had shielding

that absorbed x rays before they escaped and exposed viewers.)

show that a rock thrown from a bridge 20.0 m above water with an initial

speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent

of the direction thrown.

of the world, how many of the 9-megaton variety would be needed for a

as it may sound—there are thousands of nuclear bombs, and

their energy can be trapped in underground explosions and converted to

electricity, as natural geothermal energy is.

realized in the next century. Fusion would be a relatively clean and

illustrate this, calculate how many years the present energy needs of the

world could be supplied by one millionth of the oceans’ hydrogen fusion

energy. (b) How does this time compare with historically significant

events, such as the duration of stable economic systems?

calculate the approximate factor by which the power output of this

astronomical object has declined since its explosion.

256 CHAPTER 7 | WORK, ENERGY, AND ENERGY RESOURCES

times brighter than our entire Milky Way galaxy is the supernova? (c)

Based on your answers, discuss whether it should be possible to observe

observable galaxies, the average brightness of which is somewhat less

than our own galaxy.

power for several hours at a stretch, perhaps by pedaling a mechanism

that drives an electric generator. Neglecting any problems of generator

efficiency and practical considerations such as resting time: (a) How

many people would it take to run a 4.00-kW electric clothes dryer? (b)

How many people would it take to replace a large electric power plant

that generates 800 MW?

What is the cost of operating this air conditioner 3.00 h per day for 30.0 d

energy does this appliance consume in a year?

will it take this person to lift 2000 kg of bricks 1.50 m to a platform? (Work

done to lift his body can be omitted because it is not considered useful

output here.)

400 m (about a quarter of a mile) and encounters an average frictional

force of 1200 N. What is its average power output in watts and

horsepower if this takes 7.30 s?

40.0 hp (1 hp = 746 W) to reach a speed of 15.0 m/s, neglecting friction?

(b) How long will this acceleration take if the car also climbs a 3.00-mhigh

hill in the process?

2500-kg load a height of 35.0 m in 12.0 s, if it also increases the speed

from rest to 4.00 m/s. Note that the total mass of the counterbalanced

system is 10,000 kg—so that only 2500 kg is raised in height, but the full

10,000 kg is accelerated. (b) What does it cost, if electricity is $0.0900

operates a 2.00-W electric clock for 18 months? (b) How long can a

produce 100 MW of power to reach a speed of 250 m/s and an altitude of

12.0 km if air resistance were negligible? (b) If it actually takes 900 s,

what is the power? (c) Given this power, what is the average force of air

resistance if the airplane takes 1200 s? (Hint: You must find the distance

the plane travels in 1200 s assuming constant acceleration.)

slope at a constant 30.0 m/s while encountering wind resistance and

friction totaling 600 N. Explicitly show how you follow the steps in the

atmosphere from the Sun. (Take the power output of the Sun to be

replace an electric power plant that generates 750 MW if the collectors

convert an average of 2.00% of the maximum power into electricity. (This

small conversion efficiency is due to the devices themselves, and the fact

that the sun is directly overhead only briefly.) With the same

assumptions, what area would be needed to meet the United States’

consumption values are from 2006.)

energy in a 10.0-g pat of butter? (b) How many flights is this if each flight

has 16 stairs?

sprinter who accelerates from rest to 10.0 m/s in 3.00 s? (b) Considering

the amount of power generated, do you think a well-trained athlete could

do this repetitively for long periods of time?

who takes 1.20 s to accelerate the 7.27-kg shot from rest to 14.0 m/s,

while raising it 0.800 m. (Do not include the power produced to

accelerate his body.)

Haslam, Flickr)

(b) How many food calories would a well-conditioned athlete metabolize

in doing the same work with an efficiency of 20%?

CHAPTER 7 | WORK, ENERGY, AND ENERGY RESOURCES 257

chemical energy of body fat containing about 39 kJ/g. How many grams

of fat will you gain if you eat 10,000 kJ (about 2500 kcal) one day and do

nothing but sit relaxed for 16.0 h and sleep for the other 8.00 h? Use data

person who sleeps for 7.00 h, walks for 2.00 h, attends classes for 4.00

h, cycles for 2.00 h, sits relaxed for 3.00 h, and studies for 6.00 h.

(Studying consumes energy at the same rate as sitting in class.)

the rate of 100 W while consuming oxygen at the rate of 2.00 L/min?

such a low efficiency in this activity. Suppose a person shoveling a

footpath metabolizes food at the rate of 800 W. (a) What is her useful

power output? (b) How long will it take her to lift 3000 kg of snow 1.20 m?

(This could be the amount of heavy snow on 20 m of footpath.) (c) How

much waste heat transfer in kilojoules will she generate in the process?

some height to the ground. (a) Calculate the force produced if an 80.0-kg

person jumps from a 0.600–m-high ledge and lands stiffly, compressing

joint material 1.50 cm as a result. (Be certain to include the weight of the

person.) (b) In practice the knees bend almost involuntarily to help extend

the distance over which you stop. Calculate the force produced if the

stopping distance is 0.300 m. (c) Compare both forces with the weight of

the person.

large forces in the feet and legs. (a) Calculate the force needed to stop

the downward motion of a jogger’s leg, if his leg has a mass of 13.0 kg, a

speed of 6.00 m/s, and stops in a distance of 1.50 cm. (Be certain to

include the weight of the 75.0-kg jogger’s body.) (b) Compare this force

with the weight of the jogger.

deep knee bends in which her center of mass is lowered and raised

0.400 m. (She does work in both directions.) You may assume her

efficiency is 20%. (b) What is the average power consumption rate in

watts if she does this in 3.00 min?

flight.

backward force of 80.0 N with his arm during each 1.80 m long stroke. (a)

What is his work output in each stroke? (b) Calculate the power output of

his arms if he does 120 strokes per minute.

(a) Assuming that a mountain climber uses oxygen at twice the rate for

climbing 116 stairs per minute (because of low air temperature and

winds), calculate how many liters of oxygen a climber would need for

10.0 h of climbing. (These are liters at sea level.) Note that only 40% of

the inhaled oxygen is utilized; the rest is exhaled. (b) How much useful

work does the climber do if he and his equipment have a mass of 90.0 kg

and he gains 1000 m of altitude? (c) What is his efficiency for the 10.0-h

climb?

4500 years ago. Its square base, originally 230 m on a side, covered 13.1

pyramid’s dimensions are slightly different today due to quarrying and

some sagging.) Historians estimate that 20,000 workers spent 20 years

to construct it, working 12-hour days, 330 days per year. (a) Calculate the

gravitational potential energy stored in the pyramid, given its center of

mass is at one-fourth its height. (b) Only a fraction of the workers lifted

blocks; most were involved in support services such as building ramps

Calculate the efficiency of the workers who did the lifting, assuming there

were 1000 of them and they consumed food energy at the rate of 300

kcal/h. What does your answer imply about how much of their work went

into block-lifting, versus how much work went into friction and lifting and

lowering their own bodies? (c) Calculate the mass of food that had to be

supplied each day, assuming that the average worker required 3600 kcal

per day and that their diet was 5% protein, 60% carbohydrate, and 35%

fat. (These proportions neglect the mass of bulk and nondigestible

materials consumed.)

machines. (credit: Franck Monnier, Wikimedia Commons)

energy in a candy bar? (b) Does this seem like a long time? Discuss why

exercise is necessary but may not be sufficient to cause a person to lose

weight.

at constant speed, taking all data to be known to three digits. (b) How

much work does she do if her center of mass rises 0.240 m? (c) What is

her useful power output if she does 25 push-ups in 1 min? (Should work

258 CHAPTER 7 | WORK, ENERGY, AND ENERGY RESOURCES

done lowering her body be included? See the discussion of useful work in

exerted downward on her center of gravity (CG).

speed of 2.00 m/s and encounters air resistance of 25.0 N. Find his

power output for work done against the gravitational force and air

resistance. (b) What average force does he exert backward on the snow

to accomplish this? (c) If he continues to exert this force and to

experience the same air resistance when he reaches a level area, how

long will it take him to reach a velocity of 10.0 m/s?

of 1.25 m/s and exerts an average force of 80.0 N backward with his

arms during each 1.80 m long stroke. (a) What is his initial acceleration if

water resistance is 45.0 N? (b) What is the subsequent average

resistance force from the water during the 5.00 s it takes him to reach his

top velocity of 2.50 m/s? (c) Discuss whether water resistance seems to

increase linearly with velocity.

A toy gun uses a spring with a force constant of 300 N/m to propel a

10.0-g steel ball. If the spring is compressed 7.00 cm and friction is

negligible: (a) How much force is needed to compress the spring? (b) To

what maximum height can the ball be shot? (c) At what angles above the

horizontal may a child aim to hit a target 3.00 m away at the same height

as the gun? (d) What is the gun’s maximum range on level ground?

(a) What force must be supplied by an elevator cable to produce an

of the loaded elevator is 1500 kg? (b) How much work is done by the

cable in lifting the elevator 20.0 m? (c) What is the final speed of the

elevator if it starts from rest? (d) How much work went into thermal

energy?

A car advertisement claims that its 900-kg car accelerated from rest to

30.0 m/s and drove 100 km, gaining 3.00 km in altitude, on 1.0 gal of

gasoline. The average force of friction including air resistance was 700 N.

Assume all values are known to three significant

gases?

CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS 283

after missing the elephant?

in the problem above)? (b) What is the plane’s momentum when it is

launches these airplanes with a catapult, discuss the implications of your

answer to (b) as it relates to recoil effects of the catapult on the ship.

can have the same momentum as the truck?

to bring the car to rest.

in the combustion of gun powder. What is the average force exerted on a

0.0300-kg bullet to accelerate it to a speed of 600 m/s in a time of 2.00

ms (milliseconds)?

A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate

the force the seat belt exerts on a passenger in the car to bring him to a

halt. The mass of the passenger is 70 kg.

milliseconds from an initial speed of 4.00 m/s. (a) What is the average

force exerted on the leg, taking the effective mass of the hand and

forearm to be 1.50 kg? (b) Would the force be any different if the woman

clapped her hands together at the same speed and brought them to rest

in the same time? Explain why or why not.

A professional boxer hits his opponent with a 1000-N horizontal blow that

lasts for 0.150 s. (a) Calculate the impulse imparted by this blow. (b)

What is the opponent’s final velocity, if his mass is 105 kg and he is

motionless in midair when struck near his center of mass? (c) Calculate

the recoil velocity of the opponent’s 10.0-kg head if hit in this manner,

assuming the head does not initially transfer significant momentum to the

boxer’s body. (d) Discuss the implications of your answers for parts (b)

and (c).

Suppose a child drives a bumper car head on into the side rail, which

exerts a force of 4000 N on the car for 0.200 s. (a) What impulse is

imparted by this force? (b) Find the final velocity of the bumper car if its

initial velocity was 2.80 m/s and the car plus driver have a mass of 200

kg. You may neglect friction between the car and floor.

One hazard of space travel is debris left by previous missions. There are

several thousand objects orbiting Earth that are large enough to be

detected by radar, but there are far greater numbers of very small

objects, such as flakes of paint. Calculate the force exerted by a

0.100-mg chip of paint that strikes a spacecraft window at a relative

A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs

into a bridge abutment. (a) Calculate the average force on the person if

he is stopped by a padded dashboard that compresses an average of

1.00 cm. (b) Calculate the average force on the person if he is stopped

by an air bag that compresses an average of 15.0 cm.

Military rifles have a mechanism for reducing the recoil forces of the gun

on the person firing it. An internal part recoils over a relatively large

distance and is stopped by damping mechanisms in the gun. The larger

distance reduces the average force needed to stop the internal part. (a)

Calculate the recoil velocity of a 1.00-kg plunger that directly interacts

with a 0.0200-kg bullet fired at 600 m/s from the gun. (b) If this part is

stopped over a distance of 20.0 cm, what average force is exerted upon it

by the gun? (c) Compare this to the force exerted on the gun if the bullet

is accelerated to its velocity in 10.0 ms (milliseconds).

of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier,

and the tugboat captain’s finances. Calculate the average force exerted

on the pier using the concept of impulse. (Hint: First calculate the time it

took to bring the ship to rest.)

running at 8.00 m/s but collides head-on with a padded goalpost and

of 50.0 kg/s and a speed of 42.0 m/s. Calculate the force exerted on the

wall, assuming the water’s horizontal momentum is reduced to zero.

a nail and comes to rest after driving the nail 1.00 cm into a board. (a)

Calculate the duration of the impact. (b) What was the average force

exerted on the nail?

an equation for the kinetic energy of a particle expressed as a function of

its momentum

impulse delivered by the wall?

zero (at the highest point of a vertical toss). The racquet exerts a force of

540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using

these data, find the mass of the ball.

the horizontal. What is the impulse delivered by the foot (magnitude and

direction)?

Train cars are coupled together by being bumped into one another.

Suppose two loaded train cars are moving toward one another, the first

having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second

indicates direction of motion.) What is their final velocity?

slides on ice at a speed of 0.750 m/s. It runs into another clay model,

284 CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS

which is initially motionless and has a mass of 0.350 kg. Both being soft

clay, they naturally stick together. What is their final velocity?

70-kg passenger had collided with a car that has a mass equal to and is

traveling in the opposite direction and at the same speed? Explain your

answer.

after it hits a 150-kg deer initially running at 12.0 m/s in the same

direction? Assume the deer remains on the car.

is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and

the dove’s velocity is 7.00 m/s in the same direction?

collision in which one is initially motionless. After the collision, the moving

object is stationary and the other moves with the same speed as the

other originally had. Show that both momentum and kinetic energy are

conserved.

Two manned satellites approach one another at a relative speed of 0.250

rather than dock, what is their final relative velocity?

hockey puck slapped at him at a velocity of 35.0 m/s. Suppose the goalie

and the ice puck have an elastic collision and the puck is reflected back

in the direction from which it came. What would their final velocities be in

this case?

of a pool table and bounces straight back at 2.40 m/s (80% of its original

speed). The collision lasts 0.0150 s. (a) Calculate the average force

exerted on the ball by the bumper. (b) How much kinetic energy in joules

is lost during the collision? (c) What percent of the original energy is left?

by an initially stationary 75.0-kg skater. (a) What is their final velocity

assuming negligible friction and that the 60.0-kg skater’s original

horizontal velocity is 4.00 m/s? (b) How much kinetic energy is lost?

football player catches the ball with his feet off the ground with both of

them moving horizontally, calculate: (a) the final velocity if the ball and

player are going in the same direction and (b) the loss of kinetic energy in

this case. (c) Repeat parts (a) and (b) for the situation in which the ball

and the player are going in opposite directions. Might the loss of kinetic

energy be related to how much it hurts to catch the pass?

1100-kg artillery shell horizontally with a velocity of 575 m/s. (a) If the

shell is fired straight aft (toward the rear of the ship), there will be

negligible friction opposing the ship’s recoil. Calculate its recoil velocity.

(b) Calculate the increase in internal kinetic energy (that is, for the ship

and the shell). This energy is less than the energy released by the gun

powder—ignificant heat transfer occurs.

Two manned satellites approaching one another, at a relative speed of

(after docking) by using the frame of reference in which the first satellite

was originally at rest. (b) What is the loss of kinetic energy in this inelastic

collision? (c) Repeat both parts by using the frame of reference in which

the second satellite was originally at rest. Explain why the change in

velocity is different in the two frames, whereas the change in kinetic

energy is the same in both.

A 30,000-kg freight car is coasting at 0.850 m/s with negligible friction

under a hopper that dumps 110,000 kg of scrap metal into it. (a) What is

the final velocity of the loaded freight car? (b) How much kinetic energy is

lost?

Space probes may be separated from their launchers by exploding bolts.

(They bolt away from one another.) Suppose a 4800-kg satellite uses this

method to separate from the 1500-kg remains of its launcher, and that

5000 J of kinetic energy is supplied to the two parts. What are their

subsequent velocities using the frame of reference in which they were at

rest before separation?

3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun

loosely a few centimeters from your shoulder rather than holding it tightly

against your shoulder. (a) Calculate the recoil velocity of the rifle if it is

held loosely away from the shoulder. (b) How much kinetic energy does

the rifle gain? (c) What is the recoil velocity if the rifle is held tightly

against the shoulder, making the effective mass 28.0 kg? (d) How much

kinetic energy is transferred to the rifle-shoulder combination? The pain is

related to the amount of kinetic energy, which is significantly less in this

latter situation. (e) Calculate the momentum of a 110-kg football player

running at 8.00 m/s. Compare the player’s momentum with the

momentum of a hard-thrown 0.410-kg football that has a speed of 25.0

m/s. Discuss its relationship to this problem.

One of the waste products of a nuclear reactor is plutonium-239

⎛⎝

. This nucleus is radioactive and decays by splitting into a

, the latter

of which is also radioactive and will itself decay some time later. The

converted to kinetic energy of the helium and uranium nuclei. The mass

Calculate the velocities of the two nuclei, assuming the plutonium

nucleus is originally at rest. (b) How much kinetic energy does each

nucleus carry away? Note that the data given here are accurate to three

digits only.

The Moon’s craters are remnants of meteorite collisions. Suppose a fairly

across) strikes the Moon at a speed of 15.0 km/s. (a) At what speed does

the Moon recoil after the perfectly inelastic collision (the mass of the

collision? Such an event may have been observed by medieval English

monks who reported observing a red glow and subsequent haze about

the Moon. (c) In October 2009, NASA crashed a rocket into the Moon,

and analyzed the plume produced by the impact. (Significant amounts of

water were detected.) Answer part (a) and (b) for this real-life experiment.

The mass of the rocket was 2000 kg and its speed upon impact was

9000 km/h. How does the plume produced alter these results?

Two football players collide head-on in midair while trying to catch a

thrown football. The first player is 95.0 kg and has an initial velocity of

CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS 285

6.00 m/s, while the second player is 115 kg and has an initial velocity of

–3.50 m/s. What is their velocity just after impact if they cling together?

initially moving at 25.0 m/s just after it hits and adheres to a trash can

that is 80.0 kg and is initially at rest?

a cannon ball shot at him. The cannon ball has a mass of 10.0 kg and the

horizontal component of its velocity is 8.00 m/s when the 65.0-kg

performer catches it. If the performer is on nearly frictionless roller

skates, what is his recoil velocity?

clown throws a fake barbell away. The clown’s ice skates allow her to

recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and

the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the

barbell? (b) How much kinetic energy is gained by this maneuver? (c)

Where does the kinetic energy come from?

originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and

direction) of the second puck? (You may use the result that

masses.) (b) Confirm that the collision is elastic.

horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg

the kinetic energy of the cannon? This energy is dissipated as heat

transfer in shock absorbers that stop its recoil. (c) What happens to the

vertical component of momentum that is imparted to the cannon when it

is fired?

A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg

of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final

velocity (magnitude and direction) of the bowling ball. (b) Is the collision

elastic? (c) Linear kinetic energy is greater after the collision. Discuss

how spin on the ball might be converted to linear kinetic energy in the

collision.

Ernest Rutherford (the first New Zealander to be awarded the Nobel

Prize in Chemistry) demonstrated that nuclei were very small and dense

.

collision with a gold nucleus, calculate the helium nucleus’s final speed

and the final velocity (magnitude and direction) of the gold nucleus. (b)

What is the final kinetic energy of the helium nucleus?

Two cars collide at an icy intersection and stick together afterward. The

south. The second car has a mass of 850 kg and is approaching at

direction) of the cars. (b) How much kinetic energy is lost in the collision?

(This energy goes into deformation of the cars.) Note that because both

cars have an initial velocity, you cannot use the equations for

must look for other simplifying aspects.

stationary, prove that for an elastic collision of two objects of equal

⎞⎠

as discussed in the text.

A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a

velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless,

how far does the player recoil in the time it takes the puck to reach the

goal 15.0 m away?

Antiballistic missiles (ABMs) are designed to have very large

accelerations so that they may intercept fast-moving incoming missiles in

the short time available. What is the takeoff acceleration of a 10,000-kg

ABM that expels 196 kg of gas per second at an exhaust velocity of

What is the acceleration of a 5000-kg rocket taking off from the Moon,

expels 8.00 kg of gas per second at an exhaust velocity of

Calculate the increase in velocity of a 4000-kg space probe that expels

assume the gravitational force is negligible at the probe’s location.

Ion-propulsion rockets have been proposed for use in space. They

employ atomic ionization techniques and nuclear energy sources to

produce extremely high exhaust velocities, perhaps as great as

payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in

velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass

at the given exhaust velocity. (b) These engines are usually designed to

produce a very small thrust for a very long time—the type of engine that

might be useful on a trip to the outer planets, for example. Calculate the

velocity, assuming the acceleration due to gravity is negligible.

(a) Calculate the maximum rate at which a rocket can expel gases if its

acceleration cannot exceed seven times that of gravity. The mass of the

rocket just as it runs out of fuel is 75,000-kg, and its exhaust velocity is

. (b) Why might it be necessary to limit

the acceleration of a rocket?

experiment, calculate the average exhaust velocity of the gases expelled

from the extinguisher. Starting from rest, the final velocity is 10.0 m/s.

The total mass is initially 75.0 kg and is 70.0 kg after the extinguisher is

fired.

286 CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS

(a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity

of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done

in 0.100 s and there is a 5.00-N frictional force opposing the squid’s

movement. (b) How much energy is lost to work done against friction?

(measured horizontally) before re-entering the water. (a) Calculate the

assuming negligible lift from the air and negligible air resistance. (b) The

squid propels itself by squirting water. What fraction of its mass would it

have to eject in order to achieve the speed found in the previous part?

neglected. (c) What is unreasonable about the results? (d) Which

premise is unreasonable, or which premises are inconsistent?

Consider an astronaut in deep space cut free from her space ship and

needing to get back to it. The astronaut has a few packages that she can

throw away to move herself toward the ship. Construct a problem in

which you calculate the time it takes her to get back by throwing all the

packages at one time compared to throwing them one at a time. Among

the things to be considered are the masses involved, the force she can

exert on the packages through some distance, and the distance to the

ship.

problem in which you find the force exerted by the projectile on the plate.

Among the things to be considered are the mass and speed of the

projectile and the distance over which its speed is reduced. Your

instructor may also wish for you to consider the relative merits of

depleted uranium

55.0 N at a distance of 0.850m from the hinges. What torque are you

exerting relative to the hinges? (b) Does it matter if you push at the same

height as the hinges?

force of 165 N at a distance of 0.140 m from the center of the bolt. (a)

How much torque are you exerting in newton × meters (relative to the

center of the bolt)? (b) Convert this torque to footpounds.

horizontally and perpendicular to the door. One child pushes with a force

of 17.5 N at a distance of 0.600 m from the hinges, and the second child

pushes at a distance of 0.450 m. What force must the second child exert

to keep the door from moving? Assume friction is negligible.

example.

the seesaw 0.160 m to the left of the pivot (on the side of the lighter child)

and assuming a mass of 12.0 kg for the seesaw. The other data given in

the example remain unchanged. Explicitly show how you follow the steps

in the Problem-Solving Strategy for static equilibrium.

force exerted on the wall assuming that force is horizontal while using the

data in the schematic representation of the situation. Note that the force

exerted on the wall is equal and opposite to the force exerted on the

horse, keeping it in equilibrium. The total mass of the horse and rider is

500 kg. Take the data to be accurate to three digits.

the pivot point located at the center of the seesaw. If the children are

separated by a distance of 3 m, at what distance from the pivot point is

the small child sitting in order to maintain the balance?

mass of the horse is midway between the feet. The total mass of the

horse and rider is 500kg. (b) What is the minimum coefficient of friction

between the hooves and ground? Note that the force exerted by the wall

is horizontal.

of 20 kg and its center of gravity is at the middle of the plank, what are

without the bracing but can pivot at its base. Calculate the force exerted

by each of the 10 braces if a strong wind exerts a horizontal force of 650

N on each square meter of the wall. Assume that the net force from the

wind acts at a height halfway up the wall and that all braces exert equal

forces parallel to their lengths. Neglect the thickness of the wall.

force to the chicken’s weight? (c) Does this support the contention that

the chicken has a relatively stable construction?

entirely by its hinges and the opposite shore, so that its cables are slack.

(a) What fraction of the weight is supported by the opposite shore if the

point of support is directly beneath the cable attachments? (b) What is

the direction and magnitude of the force the hinges exert on the bridge

under these circumstances? The mass of the bridge is 2500 kg.

of mass halfway between the hinges and the cable attachments. (The

bridge is supported by the cables and hinges only.) (a) Find the force in

the cables. (b) Find the direction and magnitude of the force exerted by

the hinges on the bridge.

312 CHAPTER 9 | STATICS AND TORQUE

assuming no friction between the legs and the sidewalk. (b) What force is

exerted by each side on the hinge?

the chain breaks? (b) What force is exerted by each side on the hinge?

on each foot by the floor.

distances from it are shown.

aluminum ladder (mass 10.0 kg) against the house on a concrete pad

with the base of the ladder 2.00 m from the house. The ladder rests

against a plastic rain gutter, which we can assume to be frictionless. The

center of mass of the ladder is 2 m from the bottom. The person is

standing 3 m from the bottom. What are the magnitudes of the forces on

the ladder at the top and bottom?

from the left hand, and the hands are 0.700 m apart. Calculate the force

exerted by (a) his right hand and (b) his left hand. (c) If each hand

the one located at the center of gravity of the pole. Explicitly show how

you follow the steps in the Problem-Solving Strategy for static equilibrium

described above.

above the ground to change a tire. If you had a 2.0-m long lever, where

would you place the fulcrum if your force was limited to 300 N?

has a perpendicular lever arm of 5.50 cm, while the hands have a

perpendicular lever arm of 1.02 m? (b) What upward force should you

exert to support the wheelbarrow and its load if their combined mass is

55.0 kg? (c) What force does the wheel exert on the ground?

supporting surface? The nail puller has a mass of 2.10 kg.

the rope? (b) What force must the ceiling supply, assuming you pull

straight down on the rope? Neglect the pulley system’s mass.

assuming you pull straight up on the rope. The pulley system’s mass is

stated in the text.

gastrocnemius, and soleus muscles to calcaneus bone.

carried by a tendon over the kneecap (the patella) at the angles shown in

kneecap on the upper leg bone (the femur).

CHAPTER 9 | STATICS AND TORQUE 313

permits a person to sit, stand, and pivot.

together with a schematic representation of an equivalent lever system.

Calculate the force exerted by the upper leg muscle to lift the mass at a

constant speed. Explicitly show how you follow the steps in the Problem-

device.

major acting forces are shown. Calculate the direction and magnitude of

stationary, assuming that this force acts along a line through the center of

mass as do the weight and muscle force.

mass is not directly over the principal point of support (the atlantooccipital

joint). The muscles at the back of the neck should therefore

exert a force to keep the head erect. That is why your head falls forward

when you fall asleep in the class. (a) Calculate the force exerted by these

muscles using the information in the figure. (b) What is the force exerted

by the pivot on the head?

requiring muscle action to hold the head erect. A simplified lever system is shown.

Achilles tendon if he stands on one foot? (b) Calculate the force at the

pivot of the simplified lever system shown—that force is representative of

forces in the ankle joint.

314 CHAPTER 9 | STATICS AND TORQUE

stands on one’s toes. A simplified lever system is shown.

upper leg muscle exert to lift the child at a constant speed?

joint, enabling large forces to be exerted by the back teeth. (a) Using the

information in the figure, calculate the force exerted by the teeth on the

bullet. (b) Calculate the force on the joint.

muscle with an elastic exercise rope that obeys Hooke’s Law. Assume its

the rope is held in the hand at the same location as the book. (b) What

force is on the biceps muscle if the exercise rope is pulled straight up so

biceps muscle is still perpendicular to the forearm.

with each hand to do a push-up? Assume that she moves up at a

constant speed. (b) The triceps muscle at the back of her upper arm has

an effective lever arm of 1.75 cm, and she exerts force on the floor at a

horizontal distance of 20.0 cm from the elbow joint. Calculate the

magnitude of the force in each triceps muscle, and compare it to her

weight. (c) How much work does she do if her center of mass rises 0.240

m? (d) What is her useful power output if she does 25 pushups in one

minute?

your mother’s birthday. Your brother kicks a 500 g ball, which hits the top

of the tree at a speed of 5 m/s and stays in contact with it for 10 ms. The

ball falls to the ground near the base of the tree and the recoil of the tree

is minimal. (a) What is the force on the tree? (b) The length of the sturdy

section of the root is only 20 cm. Furthermore, the soil around the roots is

loose and we can assume that an effective force is applied at the tip of

the 20 cm length. What is the effective force exerted by the end of the tip

of the root to keep the tree from toppling? Assume the tree will be

uprooted rather than bend. (c) What could you have done to ensure that

the tree does not uproot easily?

Suppose two children are using a uniform seesaw that is 3.00 m long and

has its center of mass over the pivot. The first child has a mass of 30.0

kg and sits 1.40 m from the pivot. (a) Calculate where the second 18.0 kg

child must sit to balance the seesaw. (b) What is unreasonable about the

result? (c) Which premise is unreasonable, or which premises are

inconsistent?

Consider a method for measuring the mass of a person’s arm in

anatomical studies. The subject lies on her back, extends her relaxed

arm to the side and two scales are placed below the arm. One is placed

under the elbow and the other under the back of her hand. Construct a

problem in which you calculate the mass of the arm and find its center of

mass based on the scale readings and the distances of the scales from

the shoulder joint. You must include a free body diagram of the arm to

direct the analysis. Consider changing the position of the scale under the

hand to provide more information, if needed. You may

winds. What is its angular velocity in revolutions per second?

An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a)

acceleration of a point 9.50 cm from the axis of rotation? (c) What is the

You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and

is turning at 90.0 rpm, and you press a steel axe against it with a radial

force of 20.0 N. (a) Assuming the kinetic coefficient of friction between

steel and stone is 0.20, calculate the angular acceleration of the

grindstone. (b) How many turns will the stone make before coming to

rest?

You are told that a basketball player spins the ball with an angular

the ball starts from rest and the acceleration lasts 2.00 s? (b) What is

unreasonable about the result? (c) Which premises are unreasonable or

inconsistent?

rad/s in 0.40 s.

(a) What is its angular acceleration in rad/s2?

(b) How many revolutions does it go through in the process?

is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the

total distance traveled by the dust in 3 minutes? (Ignore accelerations

due to getting the CD rotating.)

(a) How long does it take to come to rest?

(b) How many revolutions does it make before stopping?

(a) What is the angular acceleration of its 0.280-m-radius tires, assuming

they do not slip on the pavement?

(b) How many revolutions do the tires make before coming to rest, given

(c) How long does the car take to stop completely?

(d) What distance does the car travel in this time?

(e) What was the car’s initial velocity?

(f) Do the values obtained seem reasonable, considering that this stop

happens very quickly?

engineered to enhance performance based on physical laws. (credit: Beyond Neon,

Flickr)

0.250 cm radius and that its string is being pulled.

(a) If the string is stationary and the yo-yo accelerates away from it at a

(b) What is the angular velocity after 0.750 s if it starts from rest?

(c) The outside radius of the yo-yo is 3.50 cm. What is the tangential

acceleration of a point on its edge?

does it take the father to give the merry-go-round and child an angular

velocity of 1.50 rad/s? (b) How many revolutions must he go through to

generate this velocity? (c) If he exerts a slowing force of 300 N at a

radius of 1.35 m, how long would it take him to stop them?

information. (a) The 60.0-kg skater is approximated as a cylinder that has

a 0.110-m radius. (b) The skater with arms extended is approximately a

cylinder that is 52.5 kg, has a 0.110-m radius, and has two 0.900-m-long

arms which are 3.75 kg each and extend straight out from the cylinder

like rods rotated about their ends.

an effective perpendicular lever arm of 3.00 cm, producing an angular

of the boxer’s forearm?