Essential College Physics

Andrew F. Rex, Richard Wolfson

Chapter 5

Work and Energy - all with Video Answers

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Chapter Questions

Problem 1

How much total work does gravity do on you as you climb a mountain and descend to your starting point? Given this result, why do you feel so tired when you return from such a hike?

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Problem 2

A car rounds a circular curve while its speed decreases. Is the net work done on the car positive, zero, or negative? Explain.

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Problem 3

Give an example of how the word "work" used in casual conversation differs from its meaning in physics.

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Problem 4

A factory worker pushes hard against a heavy toolbox to keep it at rest on a ramp. Is he doing work?

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Problem 5

A nonzero net force is applied to an object, but its kinetic energy doesn't change. Explain why the force must be perpendicular to the object's velocity.

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Problem 6

If an object's speed triples, by what factor does its kinetic energy increase?

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Problem 7

You drag a box $2 \mathrm{~m}$ across the floor at constant speed. Next, you drag the same box $2 \mathrm{~m}$ across the same floor, giving it constant acceleration. Compare the work done by kinetic friction in the two cases.

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Problem 8

Lift a hammer a fixed distance at constant velocity. Next, lift the same hammer the same distance with a constant upward acceleration. Compare the work you do in the two cases.

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Problem 9

Is rolling friction a conservative or nonconservative force?

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Problem 10

You launch two projectiles off a cliff at the same speed, one $30^{\circ}$ above the horizontal, the other $30^{\circ}$ below. Ignoring air resistance, compare their speeds when they hit the ground. Repeat, accounting for air resistance.

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Problem 11

Discuss whether each of these quantities can ever be negative:
(a) kinetic energy;
(b) gravitational potential energy; (c) potential energy of a spring; (d) total mechanical energy, (e) work done by the air on a projectile.

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Problem 12

Hiking trails on steep hillsides often follow zigzag paths ("switchbacks"). Use energy and power to explain their usefulness.

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Problem 13

Describe the energy transformations throughout a pole vault, from the initial run until the athlete has come to rest on the
cushion beneath the bar.

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Problem 14

A book moves $2.15 \mathrm{~m}$ in the $+x$ -direction under the influence of a $45.0-\mathrm{N}$ force, also in the $x$ -direction. The work done on the
book is (a) $20.9 \mathrm{~J} ;$ (b) $45.0 \mathrm{~J} ;$ (c) $48.4 \mathrm{~J} ;$ (d) $96.8 \mathrm{~J}$.

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Problem 15

The work done by gravity on a $0.50-\mathrm{kg}$ pro jectile that falls from $y=12.5 \mathrm{~m}$ to $y=1.5 \mathrm{~m}$ is
(a) $5.5 \mathrm{~J}$;
(b) $27 \mathrm{~J} ;$ (c) $54 \mathrm{~J} ;$ (d) $81 \mathrm{~J}$.

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Problem 16

A $0.168-\mathrm{kg}$ hockey puck slides at $11.4 \mathrm{~m} / \mathrm{s}$. The work needed to stop the puck is (a) $-21.1 \mathrm{~J}$ (b) $-12.4 \mathrm{~J}$
(c) $-10.9 \mathrm{~J} ;$ (d) $-8.3 \mathrm{~J}$.

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Problem 17

In a "dead lift," a weight lifter grabs a $185-\mathrm{kg}$ barbell and lifts it $0.550 \mathrm{~m}$ from the floor. If the barbell started and ended at rest, how much work did the weight lifter do?
(a) $997 \mathrm{~J} ;$
(b) $498 \mathrm{~J}$
(c) $249 \mathrm{~J} ;$
(d) $102 \mathrm{~J}$.

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Problem 18

A Hooke's law spring with $k=135 \mathrm{~N} / \mathrm{m}$ is compressed $9.50 \mathrm{~cm}$ from equilibrium. The work required to do this is
(a) $12.8 \mathrm{~J}$
(b) $1.22 \mathrm{~J} ;$
(c) $0.61 \mathrm{~J}$;
(d) $0.35 \mathrm{~J}$.

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Problem 19

A Hooke's law spring has $k=500 \mathrm{~N} / \mathrm{m}$. The work done in extending the spring from $x=0.30 \mathrm{~m}$ to $x=0.40 \mathrm{~m}$ is
(a) $17.5 \mathrm{~J}$
(b) $20.0 \mathrm{~J} ;$
(c) $25.0 \mathrm{~J}$
(d) $40.0 \mathrm{~J}$.

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Problem 20

A $24.5-\mathrm{kg}$ boulder that falls from a $13.4-\mathrm{m}$ cliff strikes the ground with kinetic energy
(a) $3220 \mathrm{~J}$
(b) $1610 \mathrm{~J}$;
(c) $1450 \mathrm{~J}$
(d) $328 \mathrm{~J}$.

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Problem 21

At one moment an electron is moving right with speed $v$ and kinetic energy $K$. Later, the same electron is moving left with a speed $2 v .$ Now what's its kinetic energy?
(a) $2 K ;$ (b) $-2 K$;
(c) $4 K ;$ (d) $-4 K$.

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Problem 22

A $2.15-\mathrm{kg}$ rock has kinetic energy $346 \mathrm{~J}$. After you do $-211 \mathrm{~J}$ of work on the rock, its speed is (a) $11.2 \mathrm{~m} / \mathrm{s}$;
(b) $17.9 \mathrm{~m} / \mathrm{s}$
(c) $22.8 \mathrm{~m} / \mathrm{s}$
(d) $322 \mathrm{~m} / \mathrm{s}$

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Problem 23

What's the change in potential energy of a $70-\mathrm{kg}$ mountaineer going from sea level to the 8850 -m summit of Mt. Everest?
(a) $8850 \mathrm{~J}$
(b) $6.2 \times 10^{5} \mathrm{~J}$
(c) $3.0 \times 10^{6} \mathrm{~J} ;$
(d) $6.1 \times 10^{6} \mathrm{~J}$

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Problem 24

A Hooke's law spring stores $18 \mathrm{~J}$ of energy when compressed $0.14 \mathrm{~m} .$ What's its spring constant?
(a) $1840 \mathrm{~N} / \mathrm{m}$
(b) $920 \mathrm{~N} / \mathrm{m}$
(c) $460 \mathrm{~N} / \mathrm{m} ; 120 \mathrm{~N} / \mathrm{m}$

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Problem 25

The change in kinetic energy of a $1.25-\mathrm{kg}$ projectile that rises $12.8 \mathrm{~m}$ is $(\mathrm{a})-16 \mathrm{~J} ;$ (b) $-102 \mathrm{~J} ;$ (c) $-157 \mathrm{~J} ;$ (d) $+102 \mathrm{~J}$

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Problem 26

The power required to lift a $2.85-\mathrm{kg}$ brick $10.0 \mathrm{~m}$ in $2.50 \mathrm{~s}$ is
(a) $11.4 \mathrm{~W}$;
(b) $55.9 \mathrm{~W}$ :
(c) $112 \mathrm{~W}$
(d) $147 \mathrm{~W}$.

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Problem 27

A box slides across a horizontal floor to the right, with the net force on it toward the left. Which of the following is not true?
(a) The box is slowing.
(b) The net work done on the box is negative.
(c) The work done by gravity is negative.
(d) The box will not continue to move indefinitely.

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Problem 28

If Galileo dropped a $2.50-\mathrm{kg}$ cannon ball from the $58.4-\mathrm{m}$ Tower of Pisa, how much work did gravity do on the ball?

Vishal Gupta

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Problem 29

You push a heavy box, applying a 540 -N horizontal force in the direction of motion while the box slides $3.5 \mathrm{~m}$ across the floor. How much work do you do?

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Problem 30

An object moves $2.50 \mathrm{~m}$ in the $+x$ -direction under the influence of a $125-\mathrm{N}$ force directed $50^{\circ}$ above the $x$ -axis. Find the
work done on the object.

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Problem 31

A $1320-\mathrm{kg}$ car moves in the $+x$ -direction with speed $21.5 \mathrm{~m} / \mathrm{s}$ Assuming constant braking and drag forces, find (a) the force and
(b) the work needed to stop the car in a distance of $145 \mathrm{~m}$.

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Problem 32

Arranged on the floor are five concrete blocks, each $25.0 \mathrm{~kg}$ and $0.305 \mathrm{~m}$ tall. What is the minimum work required to stack all five vertically?

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Problem 33

A $1.52-\mathrm{kg}$ book slides $1.24 \mathrm{~m}$ along a level surface. The coefficient of kinetic friction between book and surface is $0.140 .$ Find the work done by friction.

Kratika Bhadauria

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Problem 34

The book in the preceding problem is initially moving at $1.81 \mathrm{~m} / \mathrm{s}$. Find (a) the distance traveled before it stops and (b) the work done by friction in bringing the book to rest.

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Problem 35

A force $\vec{F}=2.34 \mathrm{~N} \hat{\imath}+1.06 \mathrm{~N} \hat{\jmath}$ is applied to a cement block on a level floor. Find the work done by this force if the block's displacement is
(a) $2.50 \mathrm{~m} \hat{\imath} ;$ (b) $-2.50 \mathrm{~m} \hat{\imath}$
(c) $2.50 \mathrm{~m} \hat{\imath}+2.50 \mathrm{~m} \hat{\jmath}$.

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Problem 36

$\mathrm{A}$ force $\bar{F}=13 \mathrm{~N} \hat{\imath}+13 \mathrm{~N} \hat{\jmath}$ acts on a hockey puck. Determine the work done if the force results in the puck's displacement by $4.2 \mathrm{~m}$ in the $+x$ -direction and $2.1 \mathrm{~m}$ in the $-y$ -direction.

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Problem 37

A model rocket with mass $1.85 \mathrm{~kg}$ starts from rest on the ground and accelerates upward with engine force $46.2 \mathrm{~N}$. From launch until the rocket reaches a height of $100 \mathrm{~m},$ find (a) the work done by the rocket engine, (b) the work done by gravity, and (c) the net work.

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Problem 38

A $6.1-\mathrm{kg}$ cannon ball is launched at a $45^{\circ}$ angle on level ground. The cannon muzzle is $1.8 \mathrm{~m}$ above the ground. (a) Find the work done on the ball by gravity from launch until the time it reaches the ground. (b) Repeat part (a) if the ball is launched from the edge of a $19-\mathrm{m}$ -high cliff.

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Problem 39

A $1.25-\mathrm{kg}$ block is pulled at constant speed up a frictionless $15^{\circ}$ incline with constant force $\vec{F}$ directed up the incline.
(a) Identify all forces acting on the block, and use Newton's first law to find $\vec{F}$. (b) Find the work done by $\vec{F}$ in moving the block $0.60 \mathrm{~m}$ up the incline. (c) Find the work done by gravity over the same path. (d) Combine your results to find the net work done on the block.

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Problem 40

A $45.0-\mathrm{kg}$ crate is dragged at constant velocity $8.20 \mathrm{~m}$ across a horizontal floor with a rope making a $30^{\circ}$ angle above the horizontal. The coefficient of kinetic friction is $0.250 .$ Find the work done (a) by friction and (b) by the rope.

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Problem 41

A glider (mass $m_{1}=0.15 \mathrm{~kg}$ ) on a frictionless horizontal air track is connected by a light string over a pulley to a metal block (mass $m_{2}=0.10 \mathrm{~kg}$ ) hanging vertically (Figure $\mathrm{P} 5.41$ ). The objects are released from rest and move $0.50 \mathrm{~m}$.
(a) Find the objects' acceleration. (b) Find the net work done on each.
(c) Find the work done by the string on each. (d) Find the work done by gravity on the hanging mass.

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Problem 42

In a bench press, a weight lifter presses a $105-\mathrm{kg}$ barbell $0.485 \mathrm{~m}$ straight up. If the barbell starts and ends at rest, how much work did the weight lifter do?

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Problem 43

A Hooke's law spring hangs vertically with the top end fixed. Attaching a $0.150-\mathrm{kg}$ mass to the bottom end stretches the spring $0.125 \mathrm{~m}$. (a) Find the spring constant. (b) What will be the total stretch if a $1.00-\mathrm{kg}$ mass is hung from the spring?

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Problem 44

With its double-helix structure, DNA is coiled like a spring. A biophysicist grabs the ends of a DNA strand with optical tweezers and stretches it $26 \mu \mathrm{m},$ producing 1.2 -pN tension in the strand. What's the DNA's spring constant?

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Problem 45

If $13.4 \mathrm{~J}$ of work compresses a spring $2.37 \mathrm{~cm}$, what's the spring constant?

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Problem 46

How much work does it take to compress a spring with $k=25.0 \mathrm{~N} / \mathrm{m}$ by $0.450 \mathrm{~m} ?$

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Problem 47

Find the work done in extending a spring with $k=150 \mathrm{~N} / \mathrm{m}$ from $x=0.10 \mathrm{~m}$ to $x=0.30 \mathrm{~m}$.

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Problem 48

Muscles are attached to bones by elastic bundles called tendons. For small stretches, tendons can be modeled as springs obeying Hooke's law. Experiments on the Achilles tendon found that it stretched $2.66 \mathrm{~mm}$ with a $125-\mathrm{kg}$ mass hung from it. (a) What is the spring constant of the Achilles tendon?
(b) By how much would it have to stretch to store $50.0 \mathrm{~J}$ of energy?

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Problem 49

Refer to the force-versus-position graph in Figure P5.49. How much work is done by the force for a displacement from (a) 0 to $10 \mathrm{~cm}$ (b) $5 \mathrm{~cm}$ to $10 \mathrm{~cm} ;$ (c) 0 to $15 \mathrm{~cm} ?$ (d) How much work is done by the force for a displacement from $10 \mathrm{~cm}$ to $0 \mathrm{~cm} ?$

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Matty Anderson

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Problem 50

Spider silk is one of the most remarkable elastic materials known. Consider a silk strand suspended vertically with a $0.35-\mathrm{g}$ fly stuck on the end. With the fly attached, the silk measures $28.0 \mathrm{~cm}$ in length. The resident spider, of mass $0.66 \mathrm{~g}$ senses the fly and climbs down the silk to investigate. With both spider and fly at the bottom, the silk measures $37.5 \mathrm{~cm} .$ Find
(a) the spring constant and (b) the equilibrium length of the silk.

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Problem 51

Refer to the forceversus-position graph in Figure P5.51. The force is in the $x$ -direction (positive or negative as indicated), and position is measured along the $+x$ -axis. How much work is done by the force for a displacement from (a) 0 to $2 \mathrm{~m}$ (b) $2 \mathrm{~m}$ to $3 \mathrm{~m} ;$ (c) $3 \mathrm{~m}$ to $5 \mathrm{~m}$; (d) $0 \mathrm{~m}$ to $5 \mathrm{~m} ?$ (e) How much work is done by the force for a displacement from $2 \mathrm{~m}$ to $0 ?$

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Matty Anderson

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Problem 52

Four identical springs with $k=63.4 \mathrm{kN} / \mathrm{m}$ support a car, with the car's weight distributed equally among them. Find the maximum weight for the car if the springs should be compressed no more than $4.0 \mathrm{~cm}$ when the car is at rest.

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Problem 53

How much lower does the car in the preceding problem ride with four $90-\mathrm{kg}$ passengers? $C$

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Problem 54

A force $F_{x}=4 x+12$ (in in one-dimensional motion. (a) Graph the force as a function of position.
(b) Find the work done by that force in moving an object from $x=0$ to $x=5.0 \mathrm{~m}$

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Problem 55

A spring with $k=25.0 \mathrm{~N} / \mathrm{m}$ is oriented vertically with one end fixed to the ground. A $0.100-\mathrm{kg}$ mass on top of the spring compresses it. Find the spring's maximum compression in each of these cases: (a) You hold the mass while you gently compress the spring, and when you release the mass it sits at rest atop the spring.
(b) You place the mass on the uncompressed spring and release it.
(c) You drop the mass from $10.0 \mathrm{~cm}$ above the spring.

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Problem 56

For each case below, calculate the kinetic energy of the animal described. In each case, express your answer in joules and in joules per kilogram of body mass. (a) A $62-\mathrm{kg}$ person walking at $1.0 \mathrm{~m} / \mathrm{s} ;$ (b) a $62-\mathrm{kg}$ athlete running a 4 -minute mile at constant speed; (c) a $72-\mathrm{kg}$ cheetah running at its top speed of $72 \mathrm{mph}(32 \mathrm{~m} / \mathrm{s}) ;$ (d) a $12.3 \mathrm{mg}$ froghopper that leaves the ground with initial speed $2.8 \mathrm{~m} / \mathrm{s}$.

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Problem 57

A boulder flies through the air at $12.4 \mathrm{~m} / \mathrm{s}$ with kinetic energy $305 \mathrm{~J}$. (a) What's its mass? What's the boulder's kinetic energy if its speed
(b) doubles or (c) is halved?

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Problem 58

At room temperature, a nitrogen molecule (mass = $4.65 \times 10^{-26} \mathrm{~kg}$ ) in air has kinetic energy $6.07 \times 10^{-21} \mathrm{~J}$. Find its speed.

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Problem 59

A fully loaded 737 airliner has mass $68,000 \mathrm{~kg}$.
(a) Ignoring drag, how much work do the engines need to do to achieve takeoff speed of $250 \mathrm{~km} / \mathrm{h} ?$ (b) What minimum force should the engines supply to achieve takeoff in a distance of $1.20 \mathrm{~km} ?$ (c) The 737 is powered by two engines, each of which can produce $117 \mathrm{kN}$ of force. Are they powerful enough for the takeoff of part (b)?

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Matty Anderson

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Problem 60

How much work is required to lift the aircraft of the preceding problem to its $10.5-\mathrm{km}$ cruising altitude? Compare with the work required to achieve takeoff speed.

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Problem 61

A baseball with mass $0.145 \mathrm{~kg}$ is pitched at $39.0 \mathrm{~m} / \mathrm{s}$. Upon reaching home plate, $18.4 \mathrm{~m}$ away, its speed is $36.2 \mathrm{~m} / \mathrm{s}$. If the decrease is due entirely to drag, find (a) the work done by the drag force and (b) the magnitude of the (assumed constant) drag force

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Problem 62

The Moon's mass is $7.36 \times 10^{22} \mathrm{~kg},$ and its (assumed circular) orbit has radius $3.84 \times 10^{8} \mathrm{~m}$ and a period 27.3 days. Find the Moon's kinetic energy.

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Problem 63

A $0.145-\mathrm{kg}$ baseball is struck by a bat $1.20 \mathrm{~m}$ above the ground, popping straight up at $21.8 \mathrm{~m} / \mathrm{s}$.
(a) What's the ball's kinetic energy when it leaves the bat? (b) How much work is done by gravity once the ball reaches maximum height? (c) Use your answer in part (b) to find that maximum height. (d) Find the work gravity does on the ball from when it's batted until it hits the ground. (e) Ignoring air resistance, use your answer in part
(d) to find the ball's speed at the ground.

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Matty Anderson

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Problem 64

A projectile is fired horizontally from a $35-\mathrm{m}$ cliff at $26 \mathrm{~m} / \mathrm{s} .$ Find the speed and velocity of the projectile when it strikes the ground.

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Problem 65

A rock is dropped from a 10 -m-high ledge. (a) What's its speed when it hits the ground? (b) What's its height when its speed is half the value found in part (a)?

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Problem 66

An archer fires a $0.175-\mathrm{kg}$ arrow at $27 \mathrm{~m} / \mathrm{s}$ at a $45^{\circ}$ angle. What's the arrow's kinetic energy at the moment it's fired?
(b) What's its kinetic energy at the peak of its flight?
(c) What's its peak height?

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Problem 67

A 25 -gram bullet with a speed of $310 \mathrm{~m} / \mathrm{s}$ travels $15 \mathrm{~cm}$ into a tree before stopping, what's the average force exerted to stop the bullet?

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Problem 68

A $75-\mathrm{g}$ toy rocket is launched straight up from the ground at
$19 \mathrm{~m} / \mathrm{s}$. (a) What's its kinetic energy? (b) Find the work done by gravity and the rocket's new kinetic energy after it has risen $10 \mathrm{~m}$
(c) Use your answer to part (b) to find the rocket's speed at $10 \mathrm{~m}$.

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Problem 69

A crane lifts a $750-\mathrm{kg}$ girder $8.85 \mathrm{~m}$. How much work does the crane do lifting (a) at constant speed and
(b) with upward acceleration $1.20 \mathrm{~m} / \mathrm{s}^{2} ?$

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Problem 70

The force graphed in Figure $\mathrm{P} 5.49$ is applied to a $1.8-\mathrm{kg}$ box initially at rest at $x=0$ on a frictionless, horizontal surface. Find the box's speed at
(a) $x=5 \mathrm{~cm}$
(b) $x=10 \mathrm{~cm}$
(c) $x=15 \mathrm{~cm}$

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Matty Anderson

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Problem 71

Repeat the preceding problem if the box was moving in the $+x$ -direction at $1.0 \mathrm{~m} / \mathrm{s}$ when it was at $x=0$

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Matty Anderson

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Problem 72

A $1250-\mathrm{kg}$ car going $21 \mathrm{~m} / \mathrm{s}$ has to stop suddenly. The driver locks the brakes, and the car skids to a halt in a distance of $65 \mathrm{~m}$. (a) What was the car's acceleration while stopping?
(b) How much work was done by friction to stop the car? (c) What is the coefficient of kinetic friction between tires and road?

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Problem 73

Find the change in gravitational potential energy of a $60-\mathrm{kg}$ woman climbing from sea level to the $4390-\mathrm{m}$ summit of Mt. Rainier.

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Problem 74

How far must you compress a spring with $k=650 \mathrm{~N} / \mathrm{m}$ in order to store $450 \mathrm{~J}$ of energy?

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Problem 75

A spring with $k=125 \mathrm{~N} / \mathrm{m}$ is initially compressed a distance $d=0.125 \mathrm{~m}$ from equilibrium, then it's extended the same distance from equilibrium. What's the change in potential energy?

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Problem 76

You throw a $0.13-\mathrm{kg}$ rock from a $15-\mathrm{m}$ cliff. (a) Taking zero of potential energy at the cliff top, find the rock's potential energy when first released and when it hits the ground. Then find the change in potential energy.
(b) Repeat part (a), this time taking $U=0$ at the ground. (c) Compare and discuss the results of parts (a) and (b).

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Problem 77

Energy is stored in food as the potential energy of the electrical bonds in molecules. Your body converts food energy to mechanical energy and heat. Food energy is expressed in "calories," which are actually kilocalories (kcal, with $1 \mathrm{kcal}=4.186 \mathrm{~kJ}$ ). (a) How many joules are in a 120 -kcal serving of breakfast cereal?
(b) A glass of $1 \%$ milk contains 130 kcal. How many glasses would a $62-\mathrm{kg}$ person have to drink to get the energy needed to climb a hill $125 \mathrm{~m}$ high, assuming all the milk's energy is converted to her potential energy?

Satpal Satpal

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Problem 78

(See the preceding problem.) When the body "burns" food, only about $20 \%$ of the food energy is available as mechanical energy. Suppose that a $75-\mathrm{kg}$ person consumes ice cream containing $280 \mathrm{kcal}$.
(a) How high a hill would he have to climb to "work off" those calories? (b) If he wanted to do a series of chin-ups in which he lifted his body by $50.0 \mathrm{~cm}$, how many of these would he have to do to work off the ice cream?

Satpal Satpal

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Problem 79

You're at the gym, using a weight machine to do arm raises. Each raise lifts a 20.0-N weight $45 \mathrm{~cm} .$ How many raises must you do to work off $100 \mathrm{kcal} ?$ Is this a reasonable workout session? Assume $20 \%$ conversion of food energy to mechanical energy.

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Problem 80

The total mechanical energy of an object moving at $29.2 \mathrm{~m} / \mathrm{s}$ is $563 \mathrm{~J},$ and its potential energy is $175 \mathrm{~J} .$ What's its mass?

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Problem 81

Take the ground as the zero of potential energy. (a) Find the total mechanical energy of a 45.9 -gram golf ball $23.4 \mathrm{~m}$ above the ground and moving at $31.2 \mathrm{~m} / \mathrm{s}$. (b) Ignoring drag forces, what's the ball's speed when it hits the ground?

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Problem 82

At the Zero Gravity Thrill Amusement Park near Dallas, Texas, people drop from a $30-\mathrm{m}$ tower into a net below. At what speed do they reach the net?

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Problem 83

Two men pass a $5.0-\mathrm{kg}$ "medicine ball" back and forth. (a) If one man launches the ball by pushing it from rest with a $138=\mathrm{N}$ horizontal force over $0.50 \mathrm{~m}$, how fast is the ball going when it leaves his hands? (b) How much work must the other man do to stop the ball?

Satpal Satpal

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Problem 84

(a) A horizontal spring with $k=35 \mathrm{~N} / \mathrm{m}$ is compressed $0.085 \mathrm{~m}$ and used to launch a $0.075-\mathrm{kg}$ marble.
(a) Find the marble's launch speed. (b) Repeat for a vertical launch.

Susan Hallstrom

Susan Hallstrom

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Problem 85

A roller coaster going $19.2 \mathrm{~m} / \mathrm{s}$ starts up a hill. Ignoring friction, what's its speed after it has risen $12.2 \mathrm{~m}$ vertically?

Satpal Satpal

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Problem 86

A horizontal spring with $k=75 \mathrm{~N} / \mathrm{m}$ has one end attached to a wall and the other end free. An $85-\mathrm{g}$ wad of putty is thrown horizontally at $3.4 \mathrm{~m} / \mathrm{s}$ directly toward the free end. Find the maximum spring compression.

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Numerade Educator

Problem 87

A spring with $k=1340 \mathrm{~N} / \mathrm{m}$ is oriented vertically with one end attached to the ground. A $7.27-\mathrm{kg}$ bowling ball is dropped from $1.75 \mathrm{~m}$ above the top of the spring. Find the maximum spring compression.

Satpal Satpal

Numerade Educator

Problem 88

A horizontal spring with $k=120 \mathrm{~N} / \mathrm{m}$ has one end attached to the wall. A $250-\mathrm{g}$ block is pushed onto the free end, compressing the spring by $0.150 \mathrm{~m}$. The block is then released, and the spring launches it outward. (a) Neglecting friction, what's its speed when it leaves the spring? (b) Repeat part (a) if the coefficient of kinetic friction is $0.220 .$

Satpal Satpal

Numerade Educator

Problem 89

A rubber ball is dropped from rest $2.4 \mathrm{~m}$ onto level ground.
(a) What's the ball's speed when it hits the ground?
(b) Bouncing back, the ball loses $25 \%$ of its mechanical energy. To what height does it rebound?

Satpal Satpal

Numerade Educator

Problem 90

A 4.75 -kg radio-controlled model airplane is flying $23.5 \mathrm{~m}$ above the ground with velocity $12.9 \mathrm{~m} / \mathrm{s} \hat{\imath}+3.48 \mathrm{~m} / \mathrm{s} \hat{\jmath},$ with
the $x$ -axis horizontal and the $y$ -axis vertical. (a) Taking $y=0$ at the ground, what's the plane's total mechanical energy? (b) If the engine fails and the plane plummets, what's its speed when it crashes? Neglect air resistance.

Satpal Satpal

Numerade Educator

Problem 91

A 980-kg car's parking brake fails on a $3.6^{\circ}$ incline. The coefficient of rolling friction is 0.030 , and the car rolls $35 \mathrm{~m}$ down the incline. Find the work done by (a) friction and (b) gravity. (c) Find the car's final speed.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 92

A snowboarder reaches the bottom of a frictionless "halfpipe" with speed $15.9 \mathrm{~m} / \mathrm{s} .$ The half-pipe is a half-cylinder with curvature radius $11.0 \mathrm{~m} .$ How high above the edge of the halfpipe will the snowboarder fly?

Satpal Satpal

Numerade Educator

Problem 93

$\mathrm{A}$ spring with $k=$ $42.0 \mathrm{~N} / \mathrm{m}$ is mounted horizontally at the edge of a 1.20-m-high table (Figure P5.93). The spring is compressed $5.00 \mathrm{~cm},$ and $\mathrm{a} 25.0-\mathrm{g}$
pellet is placed at its end. When the spring is released, how far (horizontally) from the edge of the table does the pellet strike the floor?

Vishal Gupta

Numerade Educator

Problem 94

A frictionless roller coaster starts from rest $25 \mathrm{~m}$ above the
ground.
(a) What's its speed when it reaches the ground?
(b) Upon reaching the ground, the track goes into a vertical, circular loop. Find the maximum loop radius such that the car will maintain contact with the track at the top.

Satpal Satpal

Numerade Educator

Problem 95

A cat jumps to a 1.15 -m-high dresser, leaving the floor at $75^{\circ}$ above the horizontal. What minimum speed must it have?

Satpal Satpal

Numerade Educator

Problem 96

A simple pendulum consists of a ball of mass $m$ attached to a light string of length $L$. The other end of the string is attached to the ceiling, so the ball swings freely in a vertical plane. The ball is pulled aside until the string makes an angle $\theta$ with the vertical, at which point the ball is released from rest. Use conservation of energy to find the ball's speed when it reaches the bottom of its arc, as a function of $L$ and $\theta .$ Evaluate numerically for $\theta=45^{\circ}$ and $L=1.20 \mathrm{~m}$

MA

Matty Anderson

Numerade Educator

Problem 97

A large spring is placed at the bottom of an elevator shaft to minimize the impact in case the elevator cable breaks. A loaded car has mass $480 \mathrm{~kg}$, and its maximum height above the spring is $11.8 \mathrm{~m}$. In order to minimize the shock, the maximum acceleration of the car after hitting the spring is $4 g .$ What should be the spring constant $k ?$

MA

Matty Anderson

Numerade Educator

Problem 98

What power is needed to lift a $350-\mathrm{kg}$ crate of bricks from the ground to the top of a 23.8 -m-high building in 1 minute?

Satpal Satpal

Numerade Educator

Problem 99

Find the work done by a motor operating at a constant $8.5 \mathrm{~kW}$ for $30 \mathrm{~s}$.

Satpal Satpal

Numerade Educator

Problem 100

A woman takes $1.2 \mathrm{~s}$ to lift a $65-\mathrm{kg}$ barbell $0.45 \mathrm{~m}$ straight up in a bench press. What's her average power output?

Satpal Satpal

Numerade Educator

Problem 101

Victoria Falls in Africa drops about $100 \mathrm{~m},$ and in the rainy season as much as 550 million $m^{3}$ of water per minute rush over the falls. What's the total power in the waterfall? Hint: The density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$

Satpal Satpal

Numerade Educator

Problem 102

Your sofa won't fit through the door of your new sixth-floor apartment, so you use a $1.12-\mathrm{kW}$ motor to lift the $86.1-\mathrm{kg}$ sofa $17.2 \mathrm{~m}$ from the street. How much time does the lift take?

Satpal Satpal

Numerade Educator

Problem 103

A motorized lift runs along a stairway inclined at $30^{\circ}$.
(a) Find the work done in lifting a $75-\mathrm{kg}$ person and $22-\mathrm{kg}$ chair if the track's length is $5.6 \mathrm{~m}$. (b) What power must the motor deliver if the person is to make it from bottom to top in $12 \mathrm{~s} ?$

Satpal Satpal

Numerade Educator

Problem 104

A 58 -kg skier is being pulled up a $12^{\circ}$ frictionless slope. What power is required for the skier to cover the entire $1.20-\mathrm{km}$ slope in $4.5 \mathrm{~min} ?$

Satpal Satpal

Numerade Educator

Problem 105

Suppose your $1320-\mathrm{kg}$ sports car has a 280 -hp engine that's $40 \%$ efficient. (That is, $40 \%$ of the 280 hp can be converted into the car's motion.) Find the car's maximum speed after accelerating from rest for $4.0 \mathrm{~s}$.

Satpal Satpal

Numerade Educator

Problem 106

A constant force $F_{x}$ acts along the $x$ -axis on an object of mass $m$ initially at rest. Find the instantaneous power delivered by that force, as a function of time.

Satpal Satpal

Numerade Educator

Problem 107

A man normally consumes 8.4 MJ of food energy per day. He then begins running a distance of $8 \mathrm{~km}$ four times per week. If he expends energy at the rate of $450 \mathrm{~W}$ while running at $12 \mathrm{~km} / \mathrm{h},$ how much more food energy should he consume daily in order to maintain constant weight?

MA

Matty Anderson

Numerade Educator

Problem 108

A 62 -kg student jogs upstairs from the first floor to the sixth, a vertical distance of $19.2 \mathrm{~m}$, in $55 \mathrm{~s}$. (a) Find the power the student expends working against gravity and compare with her average power expenditure of $100 \mathrm{~W}$.
(b) She then jogs back down to the first floor and notes that the total work she's done against gravity is zero for the round trip. Why does she still feel tired?

Satpal Satpal

Numerade Educator

Problem 109

A $0.150-\mathrm{kg}$ apple falls $2.60 \mathrm{~m}$ to the ground. (a) Find the work done by gravity. (b) Make a graph of the power supplied by gravity as a function of time over the entire fall. (c) Show that the work done by gravity is equal to the average power multiplied by the fall time

MA

Matty Anderson

Numerade Educator

Problem 110

The basal metabolic rate (BMR) measures an animal's typical resting power use. For mammals, BMR approximately obeys the equation $\mathrm{BMR} \approx A \mathrm{~m}^{3 / 4}$ (Kleiber's law), where $m$ is the mass of the animal and $A$ is a constant whose value depends on the species.
(a) What are the SI units of $A ?$ (b) According to Kleiber's law, what's the BMR of a $75-\mathrm{kg}$ person if $A=3.4$ in SI units?
(c) What's the value of $A$ for a polar bear, which has a mass of $700 \mathrm{~kg}$ and $\mathrm{BMR}=460 \mathrm{~W} ?$
(d) A $180-\mathrm{kg}$ gorilla has a BMR of $170 \mathrm{~W}$. Use Kleiber's law to predict the BMR of King Kong, a $1000-\mathrm{kg}$ gorilla, assuming $A$ is the same for all gorillas.

Brian Francisco

Brian Francisco

Numerade Educator

Problem 111

A person typically contains $5.0 \mathrm{~L}$ of blood of density $1.05 \mathrm{~g} / \mathrm{mL}$. When at rest, it normally takes $1.0 \mathrm{~min}$ to pump all this blood through the body. (a) How much work does the heart do to lift all that blood from feet to brain, a distance of $1.85 \mathrm{~m} ?$ (b) What average power does the heart expend in the process? (c) The heart's actual power consumption, for a resting person, is typically $6.0 \mathrm{~W}$. Why is this greater than the power found in part (b)? Besides the potential energy to lift the blood, where else does this power go?

Prabhu Ramji

Numerade Educator

Problem 112

According to the U.S. Department of Energy, the United States consumed about $1.03 \times 10^{20} \mathrm{~J}$ of energy in $2003 .$ Find the energy consumed in $\mathrm{kWh}$ and the cost assuming a rate of $\$ 0.12$ per $\mathrm{kWh}$.

MA

Matty Anderson

Numerade Educator

Problem 113

A 175 -lb person walking briskly on level ground at 4.5 mph consumes 7.0 kcal per minute. What distance should this person walk to "burn off" 125 kcal ( 1 kcal = $4.186 \mathrm{~kJ} \mathrm{~J} ?$

Satpal Satpal

Numerade Educator

Problem 114

Power walking on level ground for 20 min consumes 175 kcal. For a $70.0-\mathrm{kg}$ person walking at $1.5 \mathrm{~m} / \mathrm{s},$ how much food energy would be consumed in $20 \mathrm{~min}$ walking up a $10^{\circ}$ incline? (Assume $20 \%$ conversion of food energy to mechanical energy.)

Satpal Satpal

Numerade Educator

Problem 115

Figure GP5.115 shows the force that acts on an object moving along the $x$ -axis. Determine the work done as the object moves from (a) $x=0$ to $x=7.0 \mathrm{~m}$ and (b) $x=0$ to $x=12.0 \mathrm{~m}$

Satpal Satpal

Numerade Educator

Problem 116

Hair is somewhat elastic, so you can model it as an ideal spring. Experimental tests on a single strand of hair show that it stretches $2.55 \mathrm{~cm}$ when a $0.100-\mathrm{kg}$ mass is hung from it. (a) What's the spring constant of this strand? (b) How much energy is stored in it if it stretches by $15.0 \mathrm{~mm} ?$ (c) Suppose you combine 200 identical, parallel strands. What will be the spring constant of this bundle, and how much potential energy will it store if it's stretched $15.0 \mathrm{~mm} ?$

MA

Matty Anderson

Numerade Educator

Problem 117

The froghopper is the insect world's champion jumper. These insects are typically $6.1 \mathrm{~mm}$ long, have mass $12.3 \mathrm{mg},$ and leave the ground at $2.8 \mathrm{~m} / \mathrm{s}$ at $58^{\circ}$ above the horizontal. (a) How high does a froghopper go in such a leap? (b) The energy for the leap is stored in the muscles of the insect's legs, which you can model as ideal springs. If the initial compression of each of the two legs is one-third of the body length, what is their spring constant?

Satpal Satpal

Numerade Educator

Problem 118

A mass hanging from a vertical spring has gravitational potential energy, and the spring has elastic potential energy.
(a) Determine how far the spring $(k=16 \mathrm{~N} / \mathrm{m})$ stretches when a 100 -gram mass is hung from it and allowed to come to rest.
(b) If the mass is pulled down $3 \mathrm{~cm}$ further, determine the change in each type of potential energy.

Satpal Satpal

Numerade Educator

Problem 119

The force graphed in Figure $\mathrm{P} 5.51$ is applied to a $2.0-\mathrm{kg}$ block that was sliding to the right (the $+x$ -direction) over a frictionless surface with speed $5.0 \mathrm{~m} / \mathrm{s}$ at $x=0 .$ (a) Is the block ever at rest? If so, where? (b) Find a position (other than $x=0$ ) when the block is again moving to the right at $5.0 \mathrm{~m} / \mathrm{s}$.

Manish Kumar

Numerade Educator

Problem 120

Consider again Atwood's machine, described in Problem $4.62,$ in which two masses $m_{1}$ and $m_{2}$ are connected over a pulley. Assume that $m_{2}>m_{1}$. The masses are released from rest. The potential energy of $m_{2}$ decreases by $7.2 \mathrm{~J},$ while its kinetic energy increases by $3.6 \mathrm{~J} .$ The potential energy of $m_{1}$ increases by $2.4 \mathrm{~J},$ as its kinetic energy increases by $1.2 \mathrm{~J} .$ (a) Determine the net work done on the system (the two masses) by external forces. (b) What force does this work? (c) What is the ratio of the two masses? (d) Is total mechanical energy conserved?

MA

Matty Anderson

Numerade Educator

Problem 121

A $1500-\mathrm{kg}$ roller coaster (including passengers) passes point $\mathrm{A}$ at $3 \mathrm{~m} / \mathrm{s}$ (Figure GP5.121). Due to safety concerns, you must design the track so that at point $\mathrm{B}$ the passengers do not experience an upward force that exceeds $4 g .$ If the arc at $B$ is circular with radius $15 \mathrm{~m},$ (a) determine the minimum value of $h$ that satisfies this requirement, and (b) determine the speed of the coaster at $C$

Satpal Satpal

Numerade Educator

Problem 122

While driving on a straight level highway in your $1450-\mathrm{kg}$ car, you take your foot off the gas and find that your speed drops from $65 \mathrm{mi} / \mathrm{h}$ to $55 \mathrm{mi} / \mathrm{h}$ over one-tenth of a mile. Assuming your average speed during this interval was $60 \mathrm{mi} / \mathrm{h},$ find the power (in watts and horsepower) needed to keep your car moving at a constant $60 \mathrm{mi} / \mathrm{h}$.

Satpal Satpal

Numerade Educator

Problem 123

A golf ball with mass $45.9 \mathrm{~g}$ leaves the ground at $42.6 \mathrm{~m} / \mathrm{s} .$ It subsequently hits the ground at $31.9 \mathrm{~m} / \mathrm{s} .$ How much work was done by air resistance (drag)?

Satpal Satpal

Numerade Educator

Problem 124

Consider the air-track experiment shown in Figure $\mathrm{P} 5.41$ with $m_{1}=0.250 \mathrm{~kg}$ and $m_{2}=0.125 \mathrm{~kg} .$ The system is released from rest, and the hanging mass drops $0.40 \mathrm{~m}$. (a) How much work is done by gravity? (b) Use the work-energy theorem to find the speeds of both blocks. (c) Use the blocks' speeds to find their acceleration.

MA

Matty Anderson

Numerade Educator

Problem 125

A spring-loaded gun has $k=72.0 \mathrm{~N} / \mathrm{m}$. The spring is compressed $3.20 \mathrm{~cm}$ and shoots a $15-\mathrm{g}$ pellet horizontally from $1.20 \mathrm{~m}$ above the ground. (a) What's the pellet's speed when it leaves the gun? (b) What's its speed when it hits the ground? (c) How far does the pellet travel horizontally? (See Figure $\mathrm{P} 5.93 .)$

MA

Matty Anderson

Numerade Educator

Problem 126

The $36-\mathrm{kg}$ wheel of an airplane flying at $245 \mathrm{~m} / \mathrm{s}$ at an altitude of $7300 \mathrm{~m}$ falls off during a flight. (a) If the wheel hits the ground at $372 \mathrm{~m} / \mathrm{s}$, how much work was done on the wheel by air resistance (drag) during its fall?
(b) If there had been no drag, what would have been the wheel's speed when it hit the ground?

Satpal Satpal

Numerade Educator