College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 5

Energy - all with Video Answers

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Chapter Questions

Problem 1

A weight lifter lifts a $350-\mathrm{N}$ set of weights from ground level to a position over his head, a vertical distance of $2.00 \mathrm{~m}$. How much work does the weight lifter do, assuming he moves the weights at constant speed?

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

In 1990 Walter Arfeuille of Belgium lifted a $281.5-\mathrm{kg}$ object through a distance of $17.1 \mathrm{~cm}$ using only his teeth.
(a) How much work did Arfeuille do on the object?
(b) What magnitude force did he exert on the object during the lift, assuming the force was constant?

Averell Hause

Carnegie Mellon University

Problem 3

The record number of boat lifts, including the boat and its ten crew members, was achieved by Sami Heinonen and Juha Räsänen of Sweden in $2000 .$ They lifted a total mass of $653.2 \mathrm{~kg}$ approximately 4 in. off the ground a total of 24 times. Estimate the total mechanical work done by the two men in lifting the boat 24 times, assuming they applied the same force to the boat during each lift. (Neglect any work they may have done allowing the boat to drop back to the ground.)

Averell Hause

Carnegie Mellon University

Problem 4

{A}$ shopper in a supermarket pushes a cart with a force of $35 \mathrm{~N}$ directed at an angle of $25^{\circ}$ below the horizontal. The force is just sufficient to overcome various frictional forces, so the cart moves at constant speed. (a) Find the work done by the shopper as she moves down a $50.0-\mathrm{m}$ length aisle. (b) What is the net work done on the cart? Why? (c) The shopper goes down the next aisle, pushing horizontally and maintaining the same speed as before. If the work done by frictional forces doesn't change, would the shopper's applied force be larger, smaller, or the same? What about the work done on the cart by the shopper?

DW

Duane Walton

Numerade Educator

Problem 5

Starting from rest, a $5.00-\mathrm{kg}$ block slides $2.50 \mathrm{~m}$ down a rough $30.0^{\circ}$ incline. The coefficient of kinetic friction between the block and the incline is $\mu_{k}=0.436$. Determine (a) the work done by the force of gravity, (b) the work done by the friction force between block and incline, and (c) the work done by the normal force. (d) Qualitatively, how would the answers change if a shorter ramp at a steeper angle were used to span the same vertical height?

Averell Hause

Carnegie Mellon University

Problem 6

A horizontal force of $150 \mathrm{~N}$ is used to push a $40.0-\mathrm{kg}$ packing crate a distance of $6.00 \mathrm{~m}$ on a rough horizontal surface. If the crate moves at constant speed, find (a) the work done by the $150-\mathrm{N}$ force and (b) the coefficient of kinetic friction between the crate and surface.

Averell Hause

Carnegie Mellon University

Problem 7

A sledge loaded with bricks has a total mass of $18.0 \mathrm{~kg}$ and is pulled at constant speed by a rope inclined at $20.0^{\circ}$ above the horizontal. The sledge moves a distance of $20.0 \mathrm{~m}$ on a horizontal surface. The coefficient of kinetic friction between the sledge and surface is $0.500$. (a) What is the tension in the rope? (b) How much work is done by the rope on the sledge? (c) What is the mechanical energy lost due to friction?

Averell Hause

Carnegie Mellon University

Problem 8

A block of mass $2.50 \mathrm{~kg}$ is pushed $2.20 \mathrm{~m}$ along a frictionless horizontal table by a constant $16.0-\mathrm{N}$ force directed $25.0^{\circ}$ below the horizontal. Determine the work done by
(a) the applied force, (b) the normal force exerted by the table, (c) the force of gravity, and (d) the net force on the block.

Averell Hause

Carnegie Mellon University

Problem 9

A mechanic pushes a $2.50 \times 10^{3}-\mathrm{kg}$ car from rest to a speed of $v$, doing $5000 \mathrm{~J}$ of work in the process. During this time, the car moves $25.0 \mathrm{~m}$. Neglecting friction between car and road, find (a) $v$ and (b) the horizontal force exerted on the car.

Averell Hause

Carnegie Mellon University

Problem 10

A $7.00-\mathrm{kg}$ bowling ball moves at $3.00 \mathrm{~m} / \mathrm{s}$. How fast must a $2.45$ -g Ping-Pong ball move so that the two balls have the same kinetic energy?

Averell Hause

Carnegie Mellon University

Problem 11

A $5.75-\mathrm{kg}$ object is initially moving so that its $x$ -component of velocity is $6.00 \mathrm{~m} / \mathrm{s}$ and its $y$ -component of velocity is $-2.00 \mathrm{~m} / \mathrm{s}$. (a) What is the kinetic energy of the object at this time? (b) Find the change in kinetic energy of the object if its velocity changes so that its new $x$ -component is $8.50 \mathrm{~m} / \mathrm{s}$ and its new $y$ -component is $5.00 \mathrm{~m} / \mathrm{s}$.

Averell Hause

Carnegie Mellon University

Problem 12

A worker pushing a $35.0-\mathrm{kg}$ wooden crate at a constant speed for $12.0 \mathrm{~m}$ along a wood floor does $350 \mathrm{~J}$ of work by applying a constant horizontal force of magni\mathrm{\{} u d e ~ $F_{0}$ on the crate. (a) Determine the value of $F_{0}$. (b) If the worker now applies a force greater than $F_{0}$, describe the subsequent motion of the crate. (c) Describe what would happen to the crate if the applied force is less than $F_{0}$.

Averell Hause

Carnegie Mellon University

Problem 13

A 70 -kg base runner begins his slide into second base when he is moving at a speed of $4.0 \mathrm{~m} / \mathrm{s}$. The coefficient of friction between his clothes and Earth is $0.70$. He slides so that his speed is zero just as he reaches the base. (a) How much mechanical energy is lost due to friction acting on the runner? (b) How far does he slide?

Averell Hause

Carnegie Mellon University

Problem 14

An outfielder throws a $0.150-\mathrm{kg}$ baseball at a speed of $40.0 \mathrm{~m} / \mathrm{s}$ and an initial angle of $30.0^{\circ}$. What is the kinetic energy of the ball at the highest point of its motion?

Averell Hause

Carnegie Mellon University

Problem 15

A $7.80-\mathrm{g}$ bullet moving at $575 \mathrm{~m} / \mathrm{s}$ penetrates a tree trunk to a depth of $5.50 \mathrm{~cm} .$ (a) Use work and energy considerations to find the average frictional force that stops the bullet. (b) Assuming the frictional force is constant, determine how much time elapses between the moment the bullet enters the tree and the moment it stops moving.

Averell Hause

Carnegie Mellon University

Problem 16

A $0.60-\mathrm{kg}$ particle has a speed of $2.0 \mathrm{~m} / \mathrm{s}$ at point $A$ and a kinetic energy of $7.5 \mathrm{~J}$ at point $\mathrm{B}$. What is (a) its kinctic energy at A? (b) Its speed at point B? (c) The total work done on the particle as it moves from $A$ to $B$ ?

Averell Hause

Carnegie Mellon University

Problem 17

A $2000-\mathrm{kg}$ car moves down a level highway under the actions of two forces: a $1000-\mathrm{N}$ forward force exerted on the drive wheels by the road and a $950-\mathrm{N}$ resistive force. Use the work-energy theorem to find the speed of the car after it has moved a distance of $20 \mathrm{~m}$, assuming that it starts from rest.

Averell Hause

Carnegie Mellon University

Problem 18

On a frozen pond, a $10-\mathrm{kg}$ sled is given a kick that imparts to it an initial speed of $v_{0}=2.0 \mathrm{~m} / \mathrm{s} .$ The coefficient of kinetic friction between sled and ice is $\mu_{k}=0.10 .$ Use the work-energy theorem to find the distance the sled moves before coming to rest.

Averell Hause

Carnegie Mellon University

Problem 19

Find the height from which you would have to drop a ball so that it would have a speed of $9.0 \mathrm{~m} / \mathrm{s}$ just before it hits the ground.

Averell Hause

Carnegie Mellon University

Problem 20

When a $2.50-\mathrm{kg}$ object is hung vertically on a certain light spring described by Hooke's law, the spring stretches $2.76 \mathrm{~cm} .$ (a) What is the force constant of the spring?
(b) If the $2.50-\mathrm{kg}$ object is removed, how far will the spring stretch if a $1.25$ -kg block is hung on it? (c) How much work must an external agent do to stretch the same spring $8.00 \mathrm{~cm}$ from its unstretched position?

Averell Hause

Carnegie Mellon University

Problem 21

An accelerometer in a control system consists of a $3.65-\mathrm{g}$ object sliding on a horizontal rail. A low-mass spring is connected between the object and a flange at one end of the rail. Grease on the rail makes static friction negligible, but rapidly damps out vibrations of the sliding
What are the lower and upper limits of frequency to which the human ear is sensitive?

Averell Hause

Carnegie Mellon University

Problem 22

$A$ air from a trampoline with an initial speed of $9.0 \mathrm{~m} / \mathrm{s}$. The goal of this problem is to find the maximum height she attains and her speed at half maximum height. (a) What are the interacting objects and how do they interact? (b) Select the height at which the athlete's speed is $9.0 \mathrm{~m} / \mathrm{s}$ as $y=0 .$ What is her kinetic energy at this point? What is the gravitational potential energy associated with the athlete? (c) What is her kinetic energy at maximum height? What is the gravitational potential energy associated with the athlete? (d) Write a general equation for energy conservation in this case and solve for the maximum height. Substitute and obtain a numerical answer. (e) Write the general equation for energy conservation and solve for the velocity at half the maximum height. Substitute and obtain a numerical answer.

DW

Duane Walton

Numerade Educator

Problem 23

A $2300-\mathrm{kg}$ pile driver is used to drive a steel beam into the ground. The pile driver falls $7.50 \mathrm{~m}$ before coming into contacl. with the top of the beam, and it drives the beam $18.0 \mathrm{~cm}$ farther into the ground as it comes to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.

Averell Hause

Carnegie Mellon University

Problem 24

A $3.50 \times 10^{2}-\mathrm{N}$ child is in a swing that is attached to ropes $1.75 \mathrm{~m}$ long. Find the gravitational potential energy associated with the child relative to her lowest position when
(a) the ropes are horizontal, (b) the ropes make a $30.0^{\circ}$ angle with the vertical, and (c) the child is at the bottom of the circular arc.

Averell Hause

Carnegie Mellon University

Problem 25

A daredevil on a motorcycle leaves the end of a ramp with a speed of $35.0 \mathrm{~m} / \mathrm{s}$ as in Figure $\mathrm{P} 5.25 .$ If his speed is $33.0 \mathrm{~m} / \mathrm{s}$ when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.

Averell Hause

Carnegie Mellon University

Problem 26

Truck suspensions often have "helper springs" that engage at high loads. One such arrangement is a leaf spring with a helper coil spring mounted on the axle, as shown in Figure P5.26. When the main leaf spring is compressed by distance $y_{0}$, the helper spring engages and then helps Lo support any additional load. Suppose the leaf spring constant is $5.25 \times 10^{5} \mathrm{~N} / \mathrm{m}$, the helper spring constant is $3.60 \times 10^{5} \mathrm{~N} / \mathrm{m}$, and $y_{0}=0.500 \mathrm{~m} .$ (a) What is the com-
pression of the leaf spring for a load of $5.00 \times 10^{5} \mathrm{~N} ?$
(b) How much work is done in compressing the springs?

Averell Hause

Carnegie Mellon University

Problem 27

The chin-up is one exercise that can be used to strengthen the biceps muscle. This muscle can exert a force of approximately $800 \mathrm{~N}$ as it contracts a distance of $7.5 \mathrm{~cm}$ in a $75-\mathrm{kg}$ male ${ }^{3}$ How much work can the biceps muscles (one in each arm) perform in a single contraction? Compare this amount of work with the energy required to lift a $75-\mathrm{kg}$ person $40 \mathrm{~cm}$ in performing a chin-up. Do you think the biceps muscle is the only muscle involved in performing a chin-up?

Averell Hause

Carnegie Mellon University

Problem 28

A flea is able to jump about $0.5 \mathrm{~m}$. It has been said that if a flea were as big as a human, it would be able to jump over a 100 -story building! When an animal jumps, it converts work done in contracting muscles into gravita(ional potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length of contraction. If we magnified a flea by a factor of 1000, the cross section of its muscle would increase by $1000^{2}$ and the length of contraction would increase by 1000 . How high would this "superflea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)

Eduard Sanchez

Numerade Educator

Problem 29

A $50.0-\mathrm{kg}$ projectile is fired at an angle of $30.0^{\circ}$ above the horizontal with an initial speed of $1.20 \times 10^{2} \mathrm{~m} / \mathrm{s}$ from the top of a cliff $142 \mathrm{~m}$ above level ground, where the ground is taken to be $y=0 .$ (a) What is the initial total mechanical energy of the projectile? (b) Suppose the projectile is traveling $85.0 \mathrm{~m} / \mathrm{s}$ at its maximum height of $y=427 \mathrm{~m}$. How much work has been done on the projectile by air friction? (c) What is the speed of the projectile immediately before it hits the ground if air friction does one and a half times as much work on the projectile when it is going down as it did when it was going up?

Averell Hause

Carnegie Mellon University

Problem 29

A $50.0-\mathrm{kg}$ projectile is fired at an angle of $30.0^{\circ}$ above the horizontal with an initial speed of $1.20 \times 10^{2} \mathrm{~m} / \mathrm{s}$ from the top of a cliff $142 \mathrm{~m}$ above level ground, where the ground is taken to be $y=0 .$ (a) What is the initial total mechanical energy of the projectile? (b) Suppose the projectile is traveling $85.0 \mathrm{~m} / \mathrm{s}$ at its maximum height of $y=427 \mathrm{~m}$. How much work has been done on the projectile by air friction? (c) What is the speed of the projectile immediately before it hits the ground if air friction does one and a half times as much work on the projectile when it is going down as it did when it was going up?

Averell Hause

Carnegie Mellon University

Problem 30

$A$ projectile of mass $m$ is fired horizontally with an initial speed of $v_{0}$ from a height of $h$ above a flat, desert surface. Neglecting air friction, at the instant before the projectile hits the ground, find the following in terms of $m, v_{0}, h$, and $g:$ (a) the work done by the force of gravity on the projectile, (b) the change in kinetic energy of the projectile since it was fired, and (c) the final kinetic energy of the projectile. (d) Are any of the answers changed if the initial angle is changed?

Averell Hause

Carnegie Mellon University

Problem 31

A horizontal spring attached to a wall has a force constant of $850 \mathrm{~N} / \mathrm{m}$. A block of mass $1.00 \mathrm{~kg}$ is attached
to the spring and oscillates freely on a horizontal, frictionless surface as in Active Figure $5.20 .$ The initial goal of this problem is to find the velocity at the equilibrium point after the block is released. (a) What objects constitute the system, and through what forces do they interact? (b) What are the two points of interest?
(c) Find the energy stored in the spring when the mass is stretched $6.00 \mathrm{~cm}$ from equilibrium and again when the mass passes through equilibrium after being released from rest. (d) Write the conservation of energy equation fon this situation and solve it for the speed of the mass as it passes equilibrium. Substitute to obtain a numerical value. (e) What is the speed at the halfway point? Why isn't it half the speed at equilibrium?

DW

Duane Walton

Numerade Educator

Problem 32

A $50-\mathrm{kg}$ pole vaulter running at $10 \mathrm{~m} / \mathrm{s}$ vaults over the bar. Her speed when she is above the bar is $1.0 \mathrm{~m} / \mathrm{s}$. Neglect air resistance, as well as any energy absorbed by the pole, and determine her altitude as she crosses the bar.

Averell Hause

Carnegie Mellon University

Problem 33

A child and a sled with a combined mass of $50.0 \mathrm{~kg}$ slide down a frictionless slope. If the sled starts from rest and has a speed of $3.00 \mathrm{~m} / \mathrm{s}$ at the bottom, what is the height of the hill?

Averell Hause

Carnegie Mellon University

Problem 34

Hooke's law describes a certain light spring of unstretched length $35.0 \mathrm{~cm}$. When one end is attached to the top of a door frame and a $7.50-\mathrm{kg}$ object is hung from the other end, the length of the spring is $41.5 \mathrm{~cm}$. (a) Find its spring constant. (b) The load and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of $190 \mathrm{~N}$. Find the length of the spring in this situation.

Averell Hause

Carnegie Mellon University

Problem 35

A $0.250-\mathrm{kg}$ block along a horizontal track has a speed of $1.50 \mathrm{~m} / \mathrm{s}$ immediately before colliding with a light spring of force constant $4.60 \mathrm{~N} / \mathrm{m}$ located at the end of the track. (a) What is the spring's maximum compression if the track is frictionless? (b) If the track is not frictionless, would the spring's maximum compression be greater than, less than, or equal to the value obtained in $\operatorname{part}(\mathrm{a}) ?$

Averell Hause

Carnegie Mellon University

Problem 36

A bead of mass $m=5.00 \mathrm{~kg}$ is released from point (A) and slides on the frictionless track shown in Figure P5.36. Determine (a) the bead's speed at points (B) and (C) and
(b) the net work done by the force of gravity in moving the bead from (A) to (C).

Averell Hause

Carnegie Mellon University

Problem 37

Tarzan swings on a $30.0$ -m-long vine initially inclined at an angle of $37.0^{\circ}$ with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) If he pushes off with a speed of $4.00 \mathrm{~m} / \mathrm{s} ?$

Averell Hause

Carnegie Mellon University

Problem 38

A projectile is launched with a speed of $40 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ above the horizontal. Use conservation of energy to find the maximum height reached by the projectile during its flight.

Averell Hause

Carnegie Mellon University

Problem 39

The launching mechanism of a toy gun consists of a spring of unknown spring constant, as shown in Figure $\mathrm{P} 5.39 \mathrm{a}$. If the spring is compressed a distance of $0.120 \mathrm{~m}$ and the gun fired vertically as shown, the gun can launch a $20.0-\mathrm{g}$ projectile from rest to a maximum height of $20.0 \mathrm{~m}$ above the starting point of the projectile. Neglecting all resistive forces, (a) describe the mechanical energy transformations that occur from the
time the gun is fired until the projectile reaches its maximum height, (b) determine
the spring constant, and (c) find the speed of the projectile as it moves through the equilibrium position of the spring (where $x=0$ ), as shown in Figure P5.39b.

Averell Hause

Carnegie Mellon University

Problem 40

(a) A block with a mass $m$ is pulled along a horizontal surface for a distance $x$ by a constant force $\overrightarrow{\mathbf{F}}$ at an angle $\theta$ with respect to the horizontal. The coefficient of kinetic friction between block and table is $\mu_{k}$. Is the force cxerted by friction equal to $\mu_{k} m g ?$ If not, what is the force exerted by friction? (b) How much work is done by the fricuon force and by $\overrightarrow{\mathbf{F}}$ (Don't forget the signs.) (c) Identify all the forces that do no work on the block. (d) Let $m=2.00 \mathrm{~kg}, x=4.00 \mathrm{~m}, \theta=37.0^{\circ}, F=15.0 \mathrm{~N}$, and $\mu_{k}=$
$0.400$, and find the answers to parts (a) and (b).

Averell Hause

Carnegie Mellon University

Problem 41

(a) A child slides down a water slide at an amusement park from an initial height $h$. The slide can be considered frictionless because of the water flowing down it. Can the equation for conservation of mechanical energy be used on the child? (b) Is the mass of the child a factor in determining his speed at the bottom of the slide? (c) The child drops straight down rather than following the curved ramp of the slide. In which case will he be traveling faster at ground level? (d) If friction is present, how would the conservation-of-energy equation be modified? (e) Find the maximum speed of the child when the slide is fricLionless if the initial height of the slide is $12.0 \mathrm{~m}$.

Averell Hause

Carnegie Mellon University

Problem 42

An airplane of mass $1.50 \times 10^{4} \mathrm{~kg}$ is moving at $60.0 \mathrm{~m} / \mathrm{s}$. The pilot then increases the engine's thrust to $7.50 \times 10^{4} \mathrm{~N}$. The resistive force exerted by air on the airplane has a magnitude of $4.00 \times 10^{4} \mathrm{~N}$. (a) Is the work done by the engine on the airplane equal to the change in the airplane's kinetic energy after it travels through some distance through the air? Is mechanical energy conserved? Explain. (b) Find the speed of the airplane after
it has traveled $5.00 \times 10^{2} \mathrm{~m}$. Assume the airplane is in level flight throughout the motion.

Averell Hause

Carnegie Mellon University

Problem 43

A 70 -kg diver steps off a $10-\mathrm{m}$ tower and drops from rest straight down into the water. If he comes to rest $5.0 \mathrm{~m}$ beneath the surface, determine the average resistive force exerted on him by the water.

Averell Hause

Carnegie Mellon University

Problem 44

A $25,0-\mathrm{kg}$ child on a $2.00$ -m-long swing is released from rest when the ropes of the swing make an angle of $30.0^{\circ}$ with the vertical. (a) Neglecting friction, find the child's speed at the lowest position. (b) If the actual speed of the child at the lowest position is $2.00 \mathrm{~m} / \mathrm{s}$, what is the mechanical energy lost due to friction?

Averell Hause

Carnegie Mellon University

Problem 45

A $2.1 \times 10^{4}-\mathrm{kg}$ car starts from rest at the top of a $5.0$ -mlong driveway that is inclined at $20^{\circ}$ with the horizontal. If an average friction force of $4.0 \times 10^{3} \mathrm{~N}$ impedes the motion, find the speed of the car at the bottom of the driveway.

Averell Hause

Carnegie Mellon University

Problem 46

A child of mass $m$ starts from rest and slides without friction from a height $h$ along a curved waterslide (Fig. P5,46). She is launched from a height $h / 5$ into the pool.
(a) Is mechanical energy conserved? Why?
(b) Give the gravitational potential encrgy associated with the child and her kinetic energy in terms of $m g h$ at the following positions: the top of the waterslide, the launching point, and the point where she lands in the pool. (c) Determine her initial speed $v_{0}$ at the launch point in terms of $g$ and
h. (d) Determine her maximum airborne height $y_{\max }$ in terms of $h, g$, and the horizontal speed at that height, $v_{0 x^{-}}$
(c) Use the $x$ -component of the answer to part (c) to eliminate $v_{0}$ from the answer to part (d), giving the height $y_{\text {eaax }}$ in terms of $g, h$, and the launch angle $\theta$. (f) Would your answers be the same if the waterslide were not frictionless? Explain.

DW

Duane Walton

Numerade Educator

Problem 47

A skier starts from rest at the top of a hill that is inclined $10.5^{\circ}$ with respect to the horizontal. The hillside is $200 \mathrm{~m}$ long, and the coefficient of friction between snow and skis is $0.075$ 0. At the bottom of the hill, the snow is level and the coefficient of friction is unchanged. How far does the skier glide along the horizontal portion of the snow before coming to rest?

DW

Duane Walton

Numerade Educator

Problem 48

In a circus performance, a monkey is strapped to a sled and both are given an initial speed of $4.0 \mathrm{~m} / \mathrm{s}$ up a $20^{\circ}$ inclined track. The combined mass of monkey and sled is $20 \mathrm{~kg}$, and the coefficient of kinetic friction between sled and incline is $0.20 .$ How far up the incline do the monkey and sled move?

Averell Hause

Carnegie Mellon University

Problem 49

An $80.0-\mathrm{kg}$ skydiver jumps out of a balloon at an altitude of $1000 \mathrm{~m}$ and opens the parachute at an altitude of
$200.0 \mathrm{~m}$. (a) Assuming that the total retarding force on the diver is constant at $50.0 \mathrm{~N}$ with the parachute closed and constant at $3600 \mathrm{~N}$ with the parachute open, what is the speed of the diver when he lands on the ground?
(b) Do you think the skydiver will get hurt? Explain.
(c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is $5.00 \mathrm{~m} / \mathrm{s} ?$ (d) How realistic is the assumption that the total retarding force is constant? Explain.

DW

Duane Walton

Numerade Educator

Problem 50

A skier of mass $70 \mathrm{~kg}$ is pulled up a slope by a motordriven cable. (a) How much work is required to pull him $60 \mathrm{~m}$ up a $30^{\circ}$ slope (assumed frictionless) at a constant speed of $2.0 \mathrm{~m} / \mathrm{s}$ ? (b) What power must a motor have to perform this task?

Nishant Kumar

Numerade Educator

Problem 51

A $3.50-\mathrm{kN}$ piano is lifted by three workers at constant speed to an apartment $25.0 \mathrm{~m}$ above the street using a pulley system fastened to the roof of the building. Each worker is able to deliver $165 \mathrm{~W}$ of power, and the pulley system is $75.0 \%$ efficient (so that $25.0 \%$ of the mechanical energy is lost due to friction in the pulley). Neglecting the mass of the pulley, find the time required to lift the piano from the street to the apartment.

Averell Hause

Carnegie Mellon University

Problem 52

While running, a person dissipates about $0.60 \mathrm{~J}$ of mechanical energy per step per kilogram of body mass. If a 60 -kg person develops a power of $70 \mathrm{~W}$ during a race. how fast is the person running? (Assume a running step is $1.5 \mathrm{~m}$ long.

Averell Hause

Carnegie Mellon University

Problem 53

The electric motor of a model train accelerates the train from rest to $0.620 \mathrm{~m} / \mathrm{s}$ in $21.0 \mathrm{~ms}$. The total mass of the train is $875 \mathrm{~g}$. Find the average power delivered to the train during its acceleration.

Averell Hause

Carnegie Mellon University

Problem 54

When an automobile moves with constant speed down a highway, most of the power developed by the cngine is. used to compensate for the mechanical encrgy loss due to frictional forces exerted on the car by the air and the road. If the power developed by an engine is $175 \mathrm{hp}$, estimate the total frictional force acting on the car when it is moving at a speed of $29 \mathrm{~m} / \mathrm{s}$. One horsepower equals $746 \mathrm{~W}$.

Averell Hause

Carnegie Mellon University

Problem 55

An older-model car accelerates from 0 to speed $v$ in $10 \mathrm{~s}$. A newer, more powerful sports car of the same mass accelerates from 0 to $2 v$ in the same timc period. Assuming the energy coming from the engine appears only as kinetic energy of the cars, compare the power of the two cars.

Averell Hause

Carnegie Mellon University

Problem 56

A certain rain cloud at an altitude of $1.75 \mathrm{~km}$ contains $3.20 \times 10^{7} \mathrm{~kg}$ of water vapor. How long would it take for a $2.70-\mathrm{kW}$ pump to raise the same amount of water from Earth's surface to the cloud's position?

Averell Hause

Carnegie Mellon University

Problem 57

A $1.50 \times 10^{5}-\mathrm{kg}$ car starts from rest and accelerates uniformly to $18.0 \mathrm{~m} / \mathrm{s}$ in $12.0 \mathrm{~s}$. Assume that air resistance remains constant at $400 \mathrm{~N}$ during this time. Find (a) the average power developed by the engine and (b) the instantaneous power output of the engine at $t=12.0 \mathrm{~s}$, just before the car stops accelerating.

Averell Hause

Carnegie Mellon University

Problem 58

A $650-\mathrm{kg}$ elevator starts from rest and moves upward for $3.00 \mathrm{~s}$ with constant acceleration until it reaches its
cruising speed, $1.75 \mathrm{~m} / \mathrm{s} .$ (a) What is the average power of the elevator motor during this period? (b) How docs this amount of power compare with its power during an upward trip with constant speed?

Averell Hause

Carnegie Mellon University

Problem 59

The force acting on a particle varies as in Figure $\mathrm{P} 5.59 .$ Find the work done by the force as the particle mowes
(a) from $x=0$ to $x=8.00 \mathrm{~m}$, (b) from $x=8.00 \mathrm{~m}$ to $x=$ $10.0 \mathrm{~m}$, and $(\mathrm{c})$ from $x=0$ to $x=10.0 \mathrm{~m}$.

Averell Hause

Carnegie Mellon University

Problem 60

An object of mass $3.00 \mathrm{~kg}$ is subject to a force $F_{x}$ that varies with position as in Figure $\mathrm{P} 5.60$. Find the work done by the force on the object as it moves (a) from $x=0$ to $x=$ $5.00 \mathrm{~m}$, (b) from $x=5.00 \mathrm{~m}$ to $x=10.0 \mathrm{~m}$, and (c) from $x=10.0 \mathrm{~m}$ to $x=15.0 \mathrm{~m} .(\mathrm{d})$ If the object has a speed of $0.500 \mathrm{~m} / \mathrm{s}$ at $x=0$, find its speed at $x=5.00 \mathrm{~m}$ and its speed at $x=15.0 \mathrm{~m}$.

Averell Hause

Carnegie Mellon University

Problem 61

The force acting on an object is given by $F_{n}=(8 x-16) \mathrm{N}$, where $x$ is in meters. (a) Make a plot of this force versus $x$ from $x=0$ to $x=3.00 \mathrm{~m}$. (b) From your graph, find the net work done by the force as the object moves from $x=0$ (o $x=3.00 \mathrm{~m}$.

Averell Hause

Carnegie Mellon University

Problem 62

A raw egg can be dropped from a third-floor window and land on a foam-rubber pad on the ground without breaking. If a $75.0-g$ egg is dropped from a window located $92.0 \mathrm{~m}$ above the ground and a foam-rubber pad that is $15.0 \mathrm{~cm}$ thick stops the egg in $9.20 \mathrm{~ms}$, (a) by how much is the pad compressed? (b) What is the average force exerted on the egg after it strikes the pad? Note: Assume constant upward acceleration as the egg compresses the foam-rubber pad.

Averell Hause

Carnegie Mellon University

Problem 63

A person doing a chin-up weighs $700 \mathrm{~N}$, exclusive of the arms. During the first $25.0 \mathrm{~cm}$ of the lift, each arm exerts
an upward force of $355 \mathrm{~N}$ on the torso. If the upward movement starts from rest, what is the person's velocity at that point?

Averell Hause

Carnegie Mellon University

Problem 64

A boy starts at rest and slides down a frictionless slide as in Figure $\mathrm{P} 5.64$. The bottom of the track is a height $h$ above the ground. The boy then leaves the track horizontally, striking the ground a distance $d$ as shown. Using energy methods, determine the initial height $H$ of the boy in terms of $h$ and $d$

Manish Kumar

Numerade Educator

Problem 65

A roller-coaster car of mass $1.50 \times 10^{3} \mathrm{~kg}$ is initially at the top of a rise at point (A). It then moves $35.0 \mathrm{~m}$ at an angle of $50.0^{\circ}$ below the horizontal to a lower point (B). (a) Find both the potential energy of the system when the car is at points (A) and (B) and the change in potential energy as the car moves from point (a) to point (B), assuming $y=0$ at point (B). (b) Repeat part (a), this time choosing $y=0$ at point $\odot$, which is another $15.0 \mathrm{~m}$ down the same slope from point $\mathbb{B}$.

Averell Hause

Carnegie Mellon University

Problem 66

A $2.0$ -m-long pendulum is released from rest when the support string is at an angle of $25^{\circ}$ with the vertical. What is the speed of the bob at the bottom of the swing?

Averell Hause

Carnegie Mellon University

Problem 67

An archer pulls her bowstring back $0.400 \mathrm{~m}$ by exerting a force that increases uniformly from zero to $230 \mathrm{~N}$.
(a) What is the equivalent spring constant of the bow?
(b) How much work does the archer do in pulling the bow?

Averell Hause

Carnegie Mellon University

Problem 68

A block of mass $12.0 \mathrm{~kg}$ slides from rest down a frictionless $35,0^{\circ}$ incline and is stopped by a strong spring with $k=3.00 \times 10^{4} \mathrm{~N} / \mathrm{m}$. The block slides $3.00 \mathrm{~m}$ from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?

Averell Hause

Carnegie Mellon University

Problem 69

(a) A $75-\mathrm{kg}$ man steps out a window and falls (from rest) $1.0 \mathrm{~m}$ to a sidewalk. What is his speed just before his feet strike the pavement? (b) If the man falls with his knees and ankles locked, the only cushion for his fall is an approximately $0.50-\mathrm{cm}$ give in the pads of his feet. Calculate the average force exerted on him by the ground in this situation. This average force is sufficient to cause damage to cartilage in the joints or to break bones.

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Duane Walton

Numerade Educator

Problem 70

A toy gun uses a spring to project a $5.3-\mathrm{g}$ soft rubber sphere horizontally. The spring constant is $8.0 \mathrm{~N} / \mathrm{m}$, the barrel of the gun is $15 \mathrm{~cm}$ long, and a constant frictional force of $0.032 \mathrm{~N}$ exists between barrel and projectile. With what speed does the projectile leave the barrel if the spring was compressed $5.0 \mathrm{~cm}$ for this launch?

Averell Hause

Carnegie Mellon University

Problem 71

Two objects are connected by a light string passing over a light, frictionless pulley as in Figure $\mathrm{P} 5.71$. The $5.00-\mathrm{kg}$
object is released from rest at a point $4.00 \mathrm{~m}$ above the floor. (a) Determine the speed of each object when the two pass each other. (b) Determine the speed of each object $\quad w_{2}=3.0 \mathrm{c}$ at the moment the $5.00$ $\mathrm{kg}$ object hits the floor.
(c) How much higher does the $3.00-\mathrm{kg}$ object travel after the $5.00-\mathrm{kg}$ object hits the floor?

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Duane Walton

Numerade Educator

Problem 72

Two blocks, $A$ and $B$ (with mass $50 \mathrm{~kg}$ and $100 \mathrm{~kg}$, respectively), are connected by a string, as shown in Figure $\mathrm{P} 5.72 .$ The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between block $A$ and the incline is $\mu k=0.25$. Determine the change in the kinetic energy of block $A$ as it moves from (C) to (D), a distance of $20 \mathrm{~m}$ up the incline if the system starts from rest.

Ivan Kochetkov

Numerade Educator

Problem 73

IA $2.00 \times 10^{2}$ -g particle is released from rest at point $A$ on the inside of a smooth hemispherical bowl of radius $R=$ $30.0 \mathrm{~cm}$ (Fig. P5.73). Calculate (a) its gravitational poten\mathrm{\{} ~ \text {\{} i a l ~ e n e r g y ~ a t ~ $A$ relative to $B$,
(c) its speed at $B,(\mathrm{~d})$ its potential energy at $C$ relative $\mathrm{Lo}$ $B$, and (e) its kinetic energy at $C$.

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Duane Walton

Numerade Educator

Problem 74

The particle described in Problem 73 (Fig. P5.73) is released from point $A$ at rest. Its speed at $B$ is $1.50 \mathrm{~m} / \mathrm{s}$.
(a) What is its kinetic energy at $B ?$ (b) How much mechanical energy is lost as a result of friction as the particle goes from $A$ to $B ?$ (c) Is it possible to determine $\mu$ from these results in a simple manner? Explain.

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Duane Walton

Numerade Educator

Problem 75

A light spring with spring constant $1.20 \times 10^{3} \mathrm{~N} / \mathrm{m}$ hangs from an elevated support. From its lower end hangs a second light spring, which has spring constant $1.80 \times 10^{3} \mathrm{~N} / \mathrm{m} .$ A $1.50-\mathrm{kg}$ object hangs at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs, (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as being in series. Hint: Consider the forces on each spring separately:

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Duane Walton

Numerade Educator

Problem 76

Symbolic Version of Problem 75 A light spring with spring constant $k_{1}$ hangs from an elevated support. From its lower end hangs a second light spring, which has spring constant $k_{2}$. An object of mass $m$ hangs at rest from the lower end of the second spring. (a) Find the total extension distance $x$ of the pair of springs in terms of the two displacements $x_{1}$ and $x_{2} .$ (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as being in series.

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Duane Walton

Numerade Educator

Problem 77

In terms of saving energy, bicycling and walking are far more efficient means of transportation than is uravel by automobile. For example, when riding at $10.0 \mathrm{mi} / \mathrm{h}$. a cyclist. uses food energy at a rate of about $400 \mathrm{kcal} / \mathrm{h}$ above what he would use if he were merely sitting still. (In exercise physiology, power is often measured in kcal/h rather than in watts. Here, $1 \mathrm{kcal}=1$ nutritionist's Calorie $=4186 \mathrm{~J}$ ) Walking at $3.00 \mathrm{mi} / \mathrm{h}$ requires about $220 \mathrm{kcal} / \mathrm{h} . \mathrm{It}$ is interesting to compare these values with the energy consumption required for uravel by car. Gasoline yields about $1.30 \times 10^{8} \mathrm{~J} / \mathrm{gal}$. Find the fuel economy in equivalent miles per gallon for a person (a) walking

Eduard Sanchez

Numerade Educator

Problem 78

Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is 1 kilocalorie. which we define in Chapter 11 as $1 \mathrm{kcal}=4186 \mathrm{~J}$. Metabolizing 1 gram of fat can release $9.00 \mathrm{kcal}$. A student decides to try to lose weight by exercising. She plans to run up and down the stairs in a football stadium as fast as she can and as many times as necessary. Is this in itself at practical way to lose weight? To evaluate the program, suppose she runs up a flight of 80 steps, each $0.150 \mathrm{~m}$ high, in $65.0 \mathrm{~s}$. For simplicity, ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human muscles is $20,0 \%$. This means that when your body converts 100 J from metabolizing fat, 20 ] goes into doing mechanical work (here, climbing stairs). The remainder goes into internal energy. Assume the student's mass is $50.0 \mathrm{~kg}$. (a) How many times must she run the flight of stairs to lose 1 pound of fat? (b) What is her average power output, in watts and in horsepower, as she is running up the stairs?

Eduard Sanchez

Numerade Educator

Problem 79

A ski jumper starts from rest $50.0 \mathrm{~m}$ above the ground on a frictionless track and flics off the track at an angle of $45.0^{\circ}$ above the horizontal and at a height of $10.0 \mathrm{~m}$ above the level ground. Neglect air resistance. (a) What is her speed when she leaves the track? (b) What is the maximum altitucle she attains after leaving the urack?
(c) Where does she land relative to the end of the track?

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Duane Walton

Numerade Educator

Problem 80

I $5.0-\mathrm{kg}$ block is pushed $3.0 \mathrm{~m}$ up a vertical wall with constant speed by a constant force of magnitude $F$ applied at an angle of $\theta=30^{\circ}$ with the horizontal, as shown in Figure $\mathrm{P} 5.80 .$ If the coefficient of kinetic friction between block and wall is 0.30. determine the work done by (a) $\overrightarrow{\mathbf{F}}$, (b) the force of gravity, and (c) the normal force between block and wall. (d) By how much does the gravitational potential energy increase during the block's motion?

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Duane Walton

Numerade Educator

Problem 81

A child's pogo stick (Fig. P5.81) stores energy in a spring $\left(k=2.50 \times 10^{4} \mathrm{~N} / \mathrm{m}\right)$, At position $\left(x_{1}=-0.100 \mathrm{~m}\right)$
the spring compression is a maximum and the child is momentarily at rest. At position (B) $(x=0)$, the spring is relaxed and the child is moving upward. At position (C). the child is again momentarily at rest at the top of the jump. Assuming that the combined mass of child and pogo stick is $25.0 \mathrm{~kg}$, (a) calculate the total energy of the system if both potential energies are zero at $x=0$,
(b) determine $x_{\mathrm{p}}$
(c) calculate the speed of the child at $x=0,(\mathrm{~d})$ determine the value of $x$ for which the kinetic energy of the system is a maximum, and (e) obtain the child's maximum upward speed.

Eduard Sanchez

Numerade Educator

Problem 82

A hummingbird is able to hover because, as the wings move doivnwards, they exert a downward force on the air. Newton's third law tells us that the air exerts an equal and opposite force (upwards) on the wings. The average of this force must be equal to the weight of the bird when it hovers. If the wings move through a distance of $8.5 \mathrm{~cm}$ with each stroke, and the wings beat 80 times per second, determine the work performed by the wings on the air in I minute if the mass of the hummingbird is $3.0$ grams.

Eduard Sanchez

Numerade Educator

Problem 83

In the dangerous "sport" of bungee jumping, a daring student jumps from a hot-air balloon with a specially designed clastic cord attached to his waist, as shown in Figure $\mathrm{P} 5.83$. The unstretched length of the cord is $25.0 \mathrm{~m}$, the student weighs $700 \mathrm{~N}$, and the balloon is $36.0 \mathrm{~m}$ above the surface of a river below. Calculate the required force constant of the cord if the student is to stop safely $4.00 \mathrm{~m}$ above the river.

Hubert Agamasu

Numerade Educator

Problem 84

The masses of the javelin, discus, and shot are $0.80 \mathrm{~kg} .2 .0 \mathrm{~kg}$, and $7.2 \mathrm{~kg}$, respectively, and record throws in the corresponding track events are about $98 \mathrm{~m}, 74 \mathrm{~m}$, and $23 \mathrm{~m}$, respectively. Neglecting air resistance, (a) calculate the minimum initial kinetic energies that would produce these throws, and (b) estimate the average force exerted on each object during the throw, assuming the force acts over a distance of $2.0 \mathrm{~m}$. (c) Do your results suggest that air resistance is an important factor?

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Duane Walton

Numerade Educator

Problem 85

A truck travels uphill with constant velocity on a highway with a $7,0^{\circ}$ slope. A $50-\mathrm{kg}$ package sits on the floor of the back of the truck and does not slide, due to a static frictional force. During an interval in which the truck travels $340 \mathrm{~m}$, what is the net work done on the package? What is the work done on the package by the force of gravity, the normal force, and the friction force?

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Duane Walton

Numerade Educator

Problem 86

A daredevil wishes to bungee-jump from a hot-air balloon $65.0 \mathrm{~m}$ above a carnival midway (Fig. $\mathrm{P} 5.83$ ). He will use a piece of uniform elastic cord tied to a harness around his body to stop his fall at a point $10.0 \mathrm{~m}$ above the ground. Model his body as a particle and the cord as having negligible mass and a tension force described by Hooke's force law. In a preliminary test, hanging at rest from a $5.00-\mathrm{m}$ length of the cord, the jumper finds that his body weight stretches it by $1.50 \mathrm{~m}$. He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon. (a) What length of cord should he use? (b) What maximum acceleration will he experience?

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Duane Walton

Numerade Educator

Problem 87

A loaded ore car has a mass of $950 \mathrm{~kg}$ and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at $30.0^{\circ}$ above the horizontal. The car accelerates uniformly to a speed of $2.20 \mathrm{~m} / \mathrm{s}$ in $12.0 \mathrm{~s}$ and then contimues at constant speed. (a) What power must the winch motor provide when the car is moving at constant speed? (b) What maximum power must the motor provide? (c) What total energy transfers out of the motor by work by the time the car moves off the end of the track, which is of length $1250 \mathrm{~m}$ ?

Eduard Sanchez

Numerade Educator

Problem 88

An object of mass $m$ is suspended from the top of a cart by a suring of length $L$ as in Figure $\mathrm{P} 5.88 \mathrm{a}$. The cart and object are initially moving to the right at a constant speed $v_{10}$. The cart comes to rest after colliding and sticking to a bumper, as in Figure $\mathrm{P} 5.88 \mathrm{~b}$, and the suspended object swings through an angle $\theta$. (a) Show that the initial speed is $v_{0}=\sqrt{2 g L(1-\cos \theta)}$. (b) If $L=1.20 \mathrm{~m}$ and $\theta=$
$35.0^{\circ}$, find the initial speed of the cart. (Hint: The force cxerted by the string on the object does no work on the object.)

Eduard Sanchez

Numerade Educator

Problem 89

Three objects with masses $m_{1}=5.0 \mathrm{~kg}, m_{2}=10 \mathrm{~kg}$, and $m_{3}=15 \mathrm{~kg}$, respectively, are attached by strings over frictionless pulleys as indicated in Figure P5.89. The horizontal surface exerts a force of friction of $30 \mathrm{~N}$ on $m_{2}$. If the system is released from rest, use energy concepts to find the speed of $m_{3}$ after it moves down $4.0 \mathrm{~m}$.

Averell Hause

Carnegie Mellon University

Problem 90

A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral springs under tension, one near each corner of the shelf. Each tray has a mass of $580 \mathrm{~g}$ and is rectangular, $45.3 \mathrm{~cm}$ by $35.6 \mathrm{~cm}$, and $0.450 \mathrm{~cm}$ thick. (a) Show that the top tray in the stack c?n always be at the same height above the floor, however many trays are in the dispenser. (b) Find the spring constant each spring should have in order for the dispenser to function in this convenient way. Is any piece of data unnecessary for this determination?

Ma Ednelyn Lim

Numerade Educator

Problem 91

In bicycling for aerobic exercise, a woman wants her heart rate to be between 136 and 166 beats per minute. Assume that her heart rate is directly proportional to her mechanical power output. Ignore all forces on the womanplus-bicycle system, except for static friction forward on the drive wheel of the bicycle and an air resistance force proportional to the square of the bicycler's speed. When her speed is $22.0 \mathrm{~km} / \mathrm{h}$, her heart rate is $90.0$ beats per minute. In what range should her speed be so that her heart rate will be in the range she wants?

Eduard Sanchez

Numerade Educator

Problem 92

In a ncedle biopsy, a narrow strip of tissue is extracted from a patient with a hollow needle. Rather than being pushed by hand, to ensure a clean cut the needle can be fired into the patient's body by a spring. Assume the needle has mass $5.60 \mathrm{~g}$, the light spring has force constant $375 \mathrm{~N} / \mathrm{m}$, and the spring is originally compressed $8.10 \mathrm{~cm}$ to project the needle horizontally without friction. The tip of the needle then moves through $2.40 \mathrm{~cm}$ of skin and soft tissue, which exerts a resistive force of $7.60 \mathrm{~N}$ on it. Next, the needle cuts $3.50 \mathrm{~cm}$ into an organ, which exerts a backward force of $9.20 \mathrm{~N}$ on it. Find (a) the maximum speed of the needle and (b) the speed at which a flange on the back end of the needle runs into a stop, set to limit the penetration to $5.90 \mathrm{~cm}$.

Eduard Sanchez

Numerade Educator