College Physics

Hugh D. Young Philip W. Adams

Chapter 7

Work and Energy - all with Video Answers

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Chapter Questions

Problem 1

A fisherman reels in $12.0 \mathrm{~m}$ of line while landing a fish, using a constant forward pull of $25.0 \mathrm{~N}$. How much work does the tension in the line do on the fish?

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

A tennis player hits a $58.0 \mathrm{~g}$ tennis ball so that it goes straight up and reaches a maximum height of $6.17 \mathrm{~m} .$ How much work does gravity do on the ball on the way up? On the way down?

Abhishek Jana

Numerade Educator

Problem 3

A boat with a horizontal tow rope pulls a water skier. She skis off to the side, so the rope makes an angle of $15.0^{\circ}$ with the forward direction of motion. If the tension in the rope is $180 \mathrm{~N},$ how much work does the rope do on the skier during a forward displacement of $300.0 \mathrm{~m} ?$

Surjit Tewari

Numerade Educator

Problem 4

A constant horizontal pull of $8.50 \mathrm{~N}$ drags a box along a horizontal floor through a distance of $17.4 \mathrm{~m}$. (a) How much work does the pull do on the box? (b) Suppose that the same pull is exerted at an angle above the horizontal. If this pull now does $65.0 \mathrm{~J}$ of work on the box while pulling it through the same distance, what angle does the force make with the horizontal?

Prabhu Ramji

Numerade Educator

Problem 5

A rope is tied to a box and used to pull the box $1.5 \mathrm{~m}$ along a h zontal floor. The rope makes an angle of $30^{\circ}$ with the horizontal has a tension of $5 \mathrm{~N}$. The opposing friction force between the box the floor is $1 \mathrm{~N}$. How much work does each of the following forces do on the box: (a) gravity, (b) the tension in the rope, (c) friction, and
(d) the normal force? What is the total work done on the box?

Ajay Singhal

Numerade Educator

Problem 6

A $128.0 \mathrm{~N}$ carton is pulled up a frictionless baggage ramp inclined at $30.0^{\circ}$ above the horizontal by a rope exerting a $72.0 \mathrm{~N}$ pull parallel to the ramp's surface. If the carton travels $5.20 \mathrm{~m}$ along the surface of the ramp, calculate the work done on it by (a) the rope, (b) gravity, and (c) the normal force of the ramp. (d) What is the net work done on the carton? (e) Suppose that the rope is angled at $50.0^{\circ}$ above the horizontal, instead of being parallel to the ramp's surface. How much work does the rope do on the carton in this case?

Ceren Uzun

Texas Tech University

Problem 7

A factory worker moves a $30.0 \mathrm{~kg}$ crate a distance of $4.5 \mathrm{~m}$ along a level floor at constant velocity by pushing horizontally on it. The coefficient of kinetic friction between the crate and the floor is 0.25 . (a) What magnitude of force must the worker apply? (b) How much work is done on the crate by the worker's push? (c) How much work is done on the crate by friction? (d) How much work is done by the normal force? By gravity? (e) What is the net work done on the crate?

Surjit Tewari

Numerade Educator

Problem 8

II An $8.00 \mathrm{~kg}$ package in a mail-sorting room slides $2.00 \mathrm{~m}$ down a chute that is inclined at $53.0^{\circ}$ below the horizontal. The coefficient of kinetic friction between the package and the chute's surface is 0.40 . Calculate the work done on the package by (a) friction, (b) gravity, and
(c) the normal force.
(d) What is the net work done on the package?

Prabhu Ramji

Numerade Educator

Problem 9

A tow truck pulls a car $5.00 \mathrm{~km}$ along a horizontal roadway using a cable having a tension of $850 \mathrm{~N}$. (a) How much work does the cable do on the car if it pulls horizontally? If it pulls at $35.0^{\circ}$ above the horizontal? (b) How much work does the cable do on the tow truck in both cases of part (a)? (c) How much work does gravity do on the car in part (a)?

Vishal Gupta

Numerade Educator

Problem 10

II A $60 \mathrm{~kg}$ woman steps onto an up-going escalator, which has an incline of $32^{\circ}$ with respect to the horizontal and is moving at $0.5 \mathrm{~m} / \mathrm{s}$. The top of the escalator is $20 \mathrm{~m}$ above the ground level. Calculate how much work is done by (a) the friction force between the woman's feet and the escalator step, (b) gravity, and (c) the normal force on the woman's feet, as she moves from the bottom to the top of the escalator. What is the total work done on the woman as she moves from the bottom to the top?

Prabhu Ramji

Numerade Educator

Problem 11

A bullet is fired into a large stationary absorber and comes to rest. Temperature measurements of the absorber show that the bullet lost $1960 \mathrm{~J}$ of kinetic energy, and high-speed photos of the bullet show that it was moving at $965 \mathrm{~m} / \mathrm{s}$ just as it struck the absorber. What is the mass of the bullet?

Surjit Tewari

Numerade Educator

Problem 12

BIO Animal energy. Adult cheetahs, the fastest of the great cats, have a mass of about $70 \mathrm{~kg}$ and have been clocked at up to $72 \mathrm{mph}$ $(32 \mathrm{~m} / \mathrm{s}) .$ (a) How many joules of kinetic energy does such a swift cheetah have? (b) By what factor would its kinetic energy change if its speed were doubled?

Prabhu Ramji

Numerade Educator

Problem 13

A $0.145 \mathrm{~kg}$ baseball leaves a pitcher's hand at a speed of $32.0 \mathrm{~m} / \mathrm{s}$ If air drag is negligible, how much work has the pitcher done on the ball by throwing it?

Surjit Tewari

Numerade Educator

Problem 14

A $1.50 \mathrm{~kg}$ book is sliding along a rough horizontal surface. At point $A$ it is moving at $3.21 \mathrm{~m} / \mathrm{s},$ and at point $B$ it has slowed to $1.25 \mathrm{~m} / \mathrm{s}$. (a) How much work was done on the book between $A$ and $B ?$ (b) If $-0.750 \mathrm{~J}$ of work is done on the book from $B$ to $C$ how fast is it moving at point $C ?$ (c) How fast would it be moving at $C$ if $+0.750 \mathrm{~J}$ of work were done on it from $B$ to $C ?$

Paul A.

California State Polytechnic University, Pomona

Problem 15

Stopping distance of a car. The driver of an $1800 \mathrm{~kg}$ car (including passengers) traveling at $23.0 \mathrm{~m} / \mathrm{s}$ slams on the brakes, locking the wheels on the dry pavement. The coefficient of kinetic friction between rubber and dry concrete is typically $0.700 .$ (a) Use the work-energy theorem to calculate how far the car will travel before stopping. (b) How far would the car travel if it were going twice as fast? (c) What happened to the car's original kinetic energy?

Ajay Singhal

Numerade Educator

Problem 16

You throw a $20 \mathrm{~N}$ rock into the air from ground level and observe that, when it is $15.0 \mathrm{~m}$ high, it is traveling upward at $25.0 \mathrm{~m} / \mathrm{s}$. Use the work-energy theorem to find (a) the rock's speed just as it left the ground and (b) the maximum height the rock will reach.

Prabhu Ramji

Numerade Educator

Problem 17

Fleas are agile, wingless insects that feed on the blood of their hosts. Although they are typically $2-3 \mathrm{~mm}$ long with a mass of $4.5 \times 10^{-4} \mathrm{~kg},$ they have an astonishing ability to jump when threatened. Their propulsion, which can briefly produce accelerations more than 100 times that of gravity, comes not from muscles but, in fact, from an elastomeric protein called resilin, which acts as a spring. Given that the typical launch velocity of a flea is about $1 \mathrm{~m} / \mathrm{s},$ what total energy must be stored in the resilin just before the flea jumps?

Ajay Singhal

Numerade Educator

Problem 18

A $61 \mathrm{~kg}$ skier on level snow coasts $184 \mathrm{~m}$ to a stop from a speed of $12.0 \mathrm{~m} / \mathrm{s} .$ (a) Use the work-energy theorem to find the coefficient of kinetic friction between the skis and the snow. (b) Suppose a $75 \mathrm{~kg}$ skier with twice the starting speed coasted the same distance before stopping. Find the coefficient of kinetic friction between that skier's skis and the snow.

Prabhu Ramji

Numerade Educator

Problem 19

A block of ice with mass $2.00 \mathrm{~kg}$ slides $0.750 \mathrm{~m}$ down an inclined plane that slopes downward at an angle of $36.9^{\circ}$ below the horizontal. If the block of ice starts from rest, what is its final speed? You can ignore friction.

Abhishek Jana

Numerade Educator

Problem 20

To stretch a certain spring by $2.5 \mathrm{~cm}$ from its equilibrium position requires $8.0 \mathrm{~J}$ of work. (a) What is the force constant of this spring? (b) What was the maximum force required to stretch it by that distance?

Surjit Tewari

Numerade Educator

Problem 21

A spring is $17.0 \mathrm{~cm}$ long when it is lying on a table. One end is then attached to a hook and the other end is pulled by a force that increases to $25.0 \mathrm{~N}$, causing the spring to stretch to a length of $19.2 \mathrm{~cm}$. (a) What is the force constant of this spring? (b) How much work was required to stretch the spring from $17.0 \mathrm{~cm}$ to $19.2 \mathrm{~cm} ?$ (c) How long will the spring be if the $25 \mathrm{~N}$ force is replaced by a $50 \mathrm{~N}$ force?

Abhishek Jana

Numerade Educator

Problem 22

A spring with spring constant $100 \mathrm{~N} / \mathrm{m}$ and unstretched length $0.4 \mathrm{~m}$ has one end anchored to a wall and a force $F$ is applied to the other end. If the force $F$ does $200 \mathrm{~J}$ of work in stretching out the spring, (a) what is its final length, (b) what is the magnitude of $F$ at maximum elongation?

Prabhu Ramji

Numerade Educator

Problem 23

The graph in Figure 7.44 shows the magnitude of the force exerted by a given spring as a function of the distance $x$ the spring is stretched. How much work is needed to stretch this spring:
(a) a distance of $5.0 \mathrm{~cm},$ starting with it unstretched, and (b) from $x=2.0 \mathrm{~cm}$ to $x=7.0 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 24

A 575 N woman climbs a staircase that rises at $53^{\circ}$ above the horizontal and is $4.75 \mathrm{~m}$ long. Her speed is a constant $45 \mathrm{~cm} / \mathrm{s}$. (a) Is the given weight a reasonable one for an adult woman? (b) How much has the gravitational potential energy increased by her climbing the stairs?
(c) How much work has gravity done on her as she climbed the stairs?

Abhishek Jana

Numerade Educator

Problem 25

How high can we jump? The maximum height a typical human can jump from a crouched start is about $60 \mathrm{~cm} .$ By how much does the gravitational potential energy increase for a $72 \mathrm{~kg}$ person in such a jump? Where does this energy come from?

Surjit Tewari

Numerade Educator

Problem 26

A $72.0 \mathrm{~kg}$ swimmer jumps into the old swimming hole from a tree limb that is $3.25 \mathrm{~m}$ above the water. Use energy conservation to find his speed just as he hits the water (a) if he just holds his nose and drops in, (b) if he bravely jumps straight up (but just beyond the board!) at $2.50 \mathrm{~m} / \mathrm{s},$ and $(\mathrm{c})$ if he manages to jump downward at $2.50 \mathrm{~m} / \mathrm{s}$

Prabhu Ramji

Numerade Educator

Problem 27

A $2.50 \mathrm{~kg}$ mass is pushed against a horizontal spring of force constant $25.0 \mathrm{~N} / \mathrm{cm}$ on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store $11.5 \mathrm{~J}$ of potential energy, the mass is suddenly released from rest.
(a) Find the greatest speed the mass reaches. When does this occur?
(b) What is the greatest acceleration of the mass, and when does it occur?

Anand Jangid

Numerade Educator

Problem 28

A force of magnitude $800.0 \mathrm{~N}$ stretches a certain spring by $0.200 \mathrm{~m}$ from its equilibrium position. (a) What is the force constant of this spring? (b) How much elastic potential energy is stored in the spring when it is: (i) stretched $0.300 \mathrm{~m}$ from its equilibrium position and(ii) compressed by $0.300 \mathrm{~m}$ from its equilibrium position? (c) How much work was done in stretching the spring by the original $0.200 \mathrm{~m} ?$

Prabhu Ramji

Numerade Educator

Problem 29

Tendons. Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation, they obey Hooke's law. In laboratory tests on a particular tendon, it was found that, when a $250 \mathrm{~g}$ object was hung from it, the tendon stretched $1.23 \mathrm{~cm}$. (a) Find the force constant of this tendon in $\mathrm{N} / \mathrm{m}$. (b) Because of its thickness, the maximum tension this tendon can support without rupturing is $138 \mathrm{~N}$. By how much can the tendon stretch without rupturing, and how much energy is stored in it at that point?

Prabhu Ramji

Numerade Educator

Problem 30

A certain spring stores $10.0 \mathrm{~J}$ of potential energy when it is stretched by $2.00 \mathrm{~cm}$ from its equilibrium position. (a) How much potential energy would the spring store if it were stretched an additional $2.00 \mathrm{~cm} ?$ (b) How much potential energy would it store if it were compressed by $2.00 \mathrm{~cm}$ from its equilibrium position?
(c) How far from the equilibrium position would you have to stretch the string to store $20.0 \mathrm{~J}$ of potential energy? (d) What is the force constant of this spring?

Abhishek Jana

Numerade Educator

Problem 31

A $0.5 \mathrm{~kg}$ ball is thrown up into the air with an initial speed of $5 \mathrm{~m} / \mathrm{s}$ At what height does the gravitational potential energy of the ball equal its initial kinetic energy? What is the maximum height of the ball?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 32

The food calorie, equal to $4186 \mathrm{~J},$ is a measure of how much energy is released when food is metabolized by the body. A certain brand of fruit-and-cereal bar contains 140 food calories per bar. (a) If a $65 \mathrm{~kg}$ hiker eats one of these bars, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes only into increasing gravitational potential energy? (b) If, as is typical, only $20 \%$ of the food calories go into mechanical energy, what would be the answer to part (a)?

Prabhu Ramji

Numerade Educator

Problem 33

A good workout. You overindulged in a delicious dessert, so you plan to work off the extra calories at the gym. To accomplish this, you decide to do a series of arm raises while holding a $5.0 \mathrm{~kg}$ weight in one hand. The distance from your elbow to the weight is $35 \mathrm{~cm}$, and in each arm raise you start with your arm horizontal and pivot it until it is vertical. Assume that the weight of your arm is small enough compared with the weight you are lifting that you can ignore it. As is typical, your muscles are $20 \%$ efficient in converting the food energy into mechanical energy, with the rest going into heat. If your dessert contained 350 food calories, how many arm raises must you do to work off these calories? Is it realistic to do them all in one session?

Prabhu Ramji

Numerade Educator

Problem 34

BIO An exercise program. A $75 \mathrm{~kg}$ person is put on an exercise program by a physical therapist, the goal being to burn up 500 food calories in each daily session. Recall that human muscles are about $20 \%$ efficient in converting the energy they use up into mechanical energy. The exercise program consists of a set of consecutive high jumps, each one $50 \mathrm{~cm}$ into the air (which is pretty good for a human) and lasting $2.0 \mathrm{~s},$ on the average. How many jumps should the person do per session, and how much time should be set aside for each session? Do you think that this is a physically reasonable exercise session?

Prabhu Ramji

Numerade Educator

Problem 35

Tall Pacific Coast redwood trees (Sequoia sempervirens) can reach heights of about $100 \mathrm{~m}$. If air drag is negligibly small, how fast is a sequoia cone moving when it reaches the ground if it dropped from the top of a $100 \mathrm{~m}$ tree?

Abhishek Jana

Numerade Educator

Problem 36

The total height of Yosemite Falls is $2425 \mathrm{ft}$. (a) How many more joules of gravitational potential energy are there for each kilogram of water at the top of this waterfall compared with each kilogram of water at the foot of the falls? (b) Find the kinetic energy and speed of each kilogram of water as it reaches the base of the waterfall, assuming that there are no losses due to friction with the air or rocks and that the mass of water had negligible vertical speed at the top. How fast (in $\mathrm{m} / \mathrm{s}$ and $\mathrm{mph}$ ) would a $70 \mathrm{~kg}$ person have to run to have that much kinetic energy? (c) How high would Yosemite Falls have to be so that each kilogram of water at the base had twice the kinetic energy you found in part (b); twice the speed you found in part (b)?

Prabhu Ramji

Numerade Educator

Problem 37

Although the altitude may vary considerably, hailstones sometimes originate around $500 \mathrm{~m}$ (about $1500 \mathrm{ft}$ ) above the ground. (a) If we ignore air drag, how fast will these hailstones be moving when they reach the ground, assuming that they start from rest? Express your answer in $\mathrm{m} / \mathrm{s}$ and in $\mathrm{mph} .$ (b) From your own experience, are hailstones actually falling that fast when they reach the ground? Why not? What happens to most of the initial potential energy?

Ajay Singhal

Numerade Educator

Problem 38

Marbles of mass $m$ are thrown from the edge of a vertical cliff of height $h$ at speed $v_{0} .$ Neglecting air resistance, how fast (in terms of $m, h,$ and $v_{0}$ ) will these marbles be moving when they reach the bottom of the cliff if they are thrown (a) straight up, (b) straight down, or (c) horizontally away from the cliff? Will the final velocity vectors of the marbles be the same or different for each case?

Prabhu Ramji

Numerade Educator

Problem 39

a satellite of Jupiter, is the most volcanically active moon or planet in the solar system. It has volcanoes that send plumes of matter over $500 \mathrm{~km}$ high (see Figure 7.45). Due to the satellite's small mass, the acceleration due to gravity on Io is only $1.81 \mathrm{~m} / \mathrm{s}^{2},$ and Io has no appreciable atmosphere. Assume that there is no variation in gravity over the distance traveled. (a) What must be the speed of material just as it leaves the volcano to reach an altitude of $500 \mathrm{~km} ?$ (b) If the gravitational potential energy is zero at the surface, what is the potential energy for a $25 \mathrm{~kg}$ fragment at its maximum height on Io? How much would this gravitational potential energy be if it were at the same height above earth?

Prabhu Ramji

Numerade Educator

Problem 40

For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.0 -mm-long, $0.50 \mathrm{mg}$ critter can reach a height of $20 \mathrm{~cm}$ in a single leap. (a) Neglecting air drag, what is the takeoff speed of such a flea? (b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass. (c) If a $65 \mathrm{~kg}, 2.0-\mathrm{m}$ -tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could he jump, and what takeoff speed would he need? (d) In fact, most humans can jump no more than $60 \mathrm{~cm}$ from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a $65 \mathrm{~kg}$ person? (e) Where does the flea store the energy that allows it to make such a sudden leap?

Prabhu Ramji

Numerade Educator

Problem 41

A $25 \mathrm{~kg}$ child plays on a swing having support ropes that are $2.20 \mathrm{~m}$ long. A friend pulls her back until the ropes are $42^{\circ}$ from the vertical and releases her from rest. (a) What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? (b) How fast will she be moving at the bottom of the swing? (c) How much work does the tension in the ropes do as the child swings from the initial position to the bottom?

Abhishek Jana

Numerade Educator

Problem 42

II A slingshot obeying Hooke's law is used to launch pebbles vertically into the air. You observe that if you pull a pebble back $20.0 \mathrm{~cm}$ against the elastic band, the pebble goes $6.0 \mathrm{~m}$ high. (a) Assuming that air drag is negligible, how high will the pebble go if you pull it back $40.0 \mathrm{~cm}$ instead? (b) How far must you pull it back so it will reach $12.0 \mathrm{~m} ?$ (c) If you pull a pebble that is twice as heavy back $20.0 \mathrm{~cm},$ how high will it $\mathrm{go} ?$

Prabhu Ramji

Numerade Educator

Problem 43

A spring with spring constant $k$ is anchored to the wall on one side of a hockey rink. A hockey puck is pressed against the spring and then released to slide across the ice. In the process the hockey puck gains a kinetic energy $K .$ Derive an expression for the initial compression of the spring $x$ in terms of $k$ and $K$.

Prabhu Ramji

Numerade Educator

Problem 44

A $1.5 \mathrm{~kg}$ box moves back and forth on a horizontal frictionless surface between two different springs, as shown in Figure $7.46 .$ The box is initially pressed against the stronger spring, compressing it $4.0 \mathrm{~cm},$ and then is released from rest. (a) By how much will the box compress the weaker spring? (b) What is the maximum speed the box will reach?

Prabhu Ramji

Numerade Educator

Problem 45

A $12.0 \mathrm{~N}$ package of whole wheat flour is suddenly placed on the pan of a scale such as you find in grocery stores. The pan is supported from below by a vertical spring of force constant $325 \mathrm{~N} / \mathrm{m}$. If the pan has negligible weight, find the maximum distance the spring will be compressed if no energy is dissipated by friction.

Prabhu Ramji

Numerade Educator

Problem 46

A spring of negligible mass has force constant $k=1600 \mathrm{~N} / \mathrm{m}$. (a) How far must the spring be compressed so that $3.20 \mathrm{~J}$ of potential energy is stored in it? (b) You place the spring vertically with one end on the floor. You then drop a $1.20 \mathrm{~kg}$ book onto it from a height of $0.80 \mathrm{~m}$ above the top of the spring. Find the maximum distance the spring will be compressed.

Prabhu Ramji

Numerade Educator

Problem 47

A $1.50 \mathrm{~kg}$ brick is sliding along on a rough horizontal surface at $13.0 \mathrm{~m} / \mathrm{s}$. If the brick stops in $4.80 \mathrm{~s}$, how much mechanical energy is lost, and what happens to this energy?

Ajay Singhal

Numerade Educator

Problem 48

A fun-loving $11.4 \mathrm{~kg}$ otter slides up a hill and then back down to the same place. If she starts up at $5.75 \mathrm{~m} / \mathrm{s}$ and returns at $3.75 \mathrm{~m} / \mathrm{s}$ how much mechanical energy did she lose on the hill, and what happened to that energy?

Abhishek Jana

Numerade Educator

Problem 49

12.0 g plastic ball is dropped from a height of $2.50 \mathrm{~m}$ and is moving at $3.20 \mathrm{~m} / \mathrm{s}$ just before it hits the floor. How much mechanical energy was lost during the ball's fall?

Surjit Tewari

Numerade Educator

Problem 50

You are rearranging the furniture in your living room. In doing so, you push a $30 \mathrm{~kg}$ sofa to the left a total distance of $2 \mathrm{~m}$ and then you push it back to the right a distance of $1 \mathrm{~m}$. The coefficient of kinetic friction between the sofa and the hardwood floor is 0.2 . (a) What is the work done by friction in moving the sofa? (b) Suppose that you had instead simply moved the sofa to its final position by moving it directly $1 \mathrm{~m}$ to the left. What is the work done by friction in this case? (c) From the results you obtained in parts (a) and (b), what can you conclude about the nature of the friction force? Is it a conservative force? Explain.

Supratim Pal

Numerade Educator

Problem 51

While a roofer is working on a roof that slants at $36^{\circ}$ above the horizontal, he accidentally nudges his $85.0 \mathrm{~N}$ toolbox, causing it to start sliding downward, starting from rest. If it starts $4.25 \mathrm{~m}$ from the lower edge of the roof, how fast will the toolbox be moving just as it reaches the edge of the roof if the kinetic friction force on it is $22.0 \mathrm{~N} ?$

Surjit Tewari

Numerade Educator

Problem 52

A block with mass $0.50 \mathrm{~kg}$ is forced against a horizontal spring of negligible mass, compressing the spring a distance of $0.20 \mathrm{~m}$, as shown in Figure $7.47 .$ When released, the block moves on a horizontal tabletop for $1.00 \mathrm{~m}$ before coming to rest. The spring constant $k$ is $100 \mathrm{~N} / \mathrm{m} .$ What is the coefficient of kinetic friction $\mu_{\mathrm{k}}$ between the block and the tabletop?

Yaqub Khan

Numerade Educator

Problem 53

A loaded $375 \mathrm{~kg}$ toboggan is traveling on smooth horizontal snow at $4.5 \mathrm{~m} / \mathrm{s}$ when it suddenly comes to a rough region. The region is $7.0 \mathrm{~m}$ long and reduces the toboggan's speed by $1.5 \mathrm{~m} / \mathrm{s}$. (a) What average friction force did the rough region exert on the toboggan? (b) By what percent did the rough region reduce the toboggan's (i) kinetic energy and (ii) speed?

Prabhu Ramji

Numerade Educator

Problem 54

A $62.0 \mathrm{~kg}$ skier is moving at $6.50 \mathrm{~m} / \mathrm{s}$ on a frictionless, horizontal snow-covered plateau when she encounters a rough patch $3.50 \mathrm{~m}$ long. The coefficient of kinetic friction between this patch and her skis is 0.300 . After crossing the rough patch and returning to friction-free snow, she skis down an icy, frictionless hill $2.50 \mathrm{~m}$ high. (a) How fast is the skier moving when she gets to the bottom of the hill? (b) How much internal energy was generated in crossing the rough patch?

Max Jennings

Numerade Educator

Problem 55

Suppose you were to drop a 14 lb bowling ball from the top of the Empire State Building, which is about $440 \mathrm{~m}$ tall, onto a machine that would catch it and then convert its kinetic energy into electrical energy. For how long could the resulting energy light a $100 \mathrm{~W}$ lightbulb? Is it actually possible to convert mechanical energy to electrical energy with $100 \%$ efficiency?

Ajay Singhal

Numerade Educator

Problem 56

The engine of a motorboat delivers $30.0 \mathrm{~kW}$ to the propeller while the boat is moving at $15.0 \mathrm{~m} / \mathrm{s}$. What would be the tension in the towline if the boat were being towed at the same speed?

Prabhu Ramji

Numerade Educator

Problem 57

At 7.35 cents per kilowatt-hour, (a) what does it cost to operate a 10.0 hp motor for $8.00 \mathrm{~h} ?$ (b) What does it cost to leave a $75 \mathrm{~W}$ light burning 24 hours a day?

Ajay Singhal

Numerade Educator

Problem 58

A tandem (two-person) bicycle team must overcome a force of $165 \mathrm{~N}$ to maintain a speed of $9.00 \mathrm{~m} / \mathrm{s}$. Find the power required per rider, assuming that each contributes equally.

Abhishek Jana

Numerade Educator

Problem 59

An elevator has mass $600 \mathrm{~kg}$, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of $20.0 \mathrm{~m}$ (five floors) in $16.0 \mathrm{~s}$, and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass $65.0 \mathrm{~kg}$.

Surjit Tewari

Numerade Educator

Problem 60

U.S. power use. The total consumption of electrical energy in the United States is about $1.0 \times 10^{19}$ joules per year. (a) Express this rate in watts and kilowatts. (b) If the U.S. population is about 320 million people, what is the average rate of electrical energy consumption per person?

Prabhu Ramji

Numerade Educator

Problem 61

Solar energy. The sun transfers energy to the earth by radiation at a rate of approximately $1.0 \mathrm{~kW}$ per square meter of surface.
(a) If this energy could be collected and converted to electrical energy with $25 \%$ efficiency, how large an area (in square kilometers) would be required to collect the electrical energy used by the United States? (See the previous problem.) (b) If the solar collectors were arranged in a square array, what would be the length of its sides in kilometers and in miles? Does an array of these dimensions seem technologically feasible?

Prabhu Ramji

Numerade Educator

Problem 62

A $20.0 \mathrm{~kg}$ box is pulled along a rough horizontal surface with a rope at a constant speed of $8 \mathrm{~m} / \mathrm{s}$. The rope is parallel to the floor, and the coefficient of kinetic friction between the box and the floor is 0.200 . What
power must the rope supply to the box? Where does this energy go?

Prabhu Ramji

Numerade Educator

Problem 63

BIO A typical flying insect applies an average force equal to twice its weight during each downward stroke while hovering. Take the mass of the insect to be $10 \mathrm{~g},$ and assume the wings move an average downward distance of $1.0 \mathrm{~cm}$ during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.

Prabhu Ramji

Numerade Educator

Problem 64

When its $75 \mathrm{~kW}$ ( $100 \mathrm{hp}$ ) engine is generating full power, a small single-engine airplane with mass $700 \mathrm{~kg}$ gains altitude at a rate of $2.5 \mathrm{~m} / \mathrm{s}(150 \mathrm{~m} / \mathrm{min},$ or $500 \mathrm{ft} / \mathrm{min}) .$ What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.)

Paul A.

California State Polytechnic University, Pomona

Problem 65

The power of the human heart. The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about $7500 \mathrm{~L}$ of blood. Assume that the work done by the heart is equal to the work required to lift that amount of blood a height equal to that of the average American female, approximately $1.63 \mathrm{~m} .$ The density of blood is $1050 \mathrm{~kg} / \mathrm{m}^{3} .$ (a) How much work does the heart do in a day? (b) What is the heart's power output in watts? (c) In fact, the heart puts out more power than you found in part (b). Why? What other forms of energy does it give the blood?

Prabhu Ramji

Numerade Educator

Problem 66

At the site of a wind farm in North Dakota, the average wind speed is $9.3 \mathrm{~m} / \mathrm{s},$ and the average density of air is $1.2 \mathrm{~kg} / \mathrm{m}^{3}$
(a) Calculate how much kinetic energy the wind contains, per cubic meter, at this location. (b) No wind turbine can capture all of the energy contained in the wind, the main reason being that capturing all the energy would require stopping the wind completely, meaning that air would stop flowing through the turbine. Suppose a particular turbine has blades with a radius of $41 \mathrm{~m}$ and is able to capture $35 \%$ of the available wind energy. What would be the power output of this turbine, under average wind conditions?

Prabhu Ramji

Numerade Educator

Problem 67

DATA A physics student measures the energy stored in a spring as a function of the distance it is stretched beyond its undistorted length. Her data are given in the table.
$$
\begin{array}{cc}
x(\mathrm{~cm}) & \text { Energy (J) } \\
\hline 2.6 & 0.34 \\
6.3 & 2.00 \\
7.5 & 2.81 \\
8.2 & 3.36
\end{array}
$$
Draw a linearized graph of the data by plotting the spring's energy as a function of the square of the distance it is stretched. Using a linear "best fit" to the data, determine the force constant of the spring.

David González Cornejo

David González Cornejo

Numerade Educator

Problem 68

Human terminal velocity. By landing properly and on soft ground (and by being lucky!), humans have survived falls from airplanes when, for example, a parachute failed to open, with astonishingly little injury. Without a parachute, a typical human eventually reaches a terminal velocity of about $62 \mathrm{~m} / \mathrm{s}$. Suppose the fall is from an airplane $1000 \mathrm{~m}$ high. (a) How fast would a person be falling when he reached the ground if there were no air drag? (b) If a $70 \mathrm{~kg}$ person reaches the ground traveling at the terminal velocity of $62 \mathrm{~m} / \mathrm{s},$ how much mechanical energy was lost during the fall? What happened to that energy?

Prabhu Ramji

Numerade Educator

Problem 69

A wooden rod of negligible mass and length $80.0 \mathrm{~cm}$ is pivoted about a horizontal axis through its center. A white rat with mass $0.500 \mathrm{~kg}$ clings to one end of the stick, and a mouse with mass $0.200 \mathrm{~kg}$ clings to the other end. The system is released from rest with the rod horizontal. If the animals can manage to hold on, what are their speeds as the rod swings through a vertical position?

Surjit Tewari

Numerade Educator

Problem 70

Ski jump ramp. You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height $h$ from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of $2.0 \mathrm{~m} / \mathrm{s}$ as they reach the gate. For safety, the skiers should have a speed of no more than $30.0 \mathrm{~m} / \mathrm{s}$ when they reach the bottom of the ramp. You determine that for an $85.0 \mathrm{~kg}$ skier with good form, friction and air resistance will do total work of magnitude $4000 \mathrm{~J}$ on him during his run down the slope. What is the maximum height $h$ for which the maximum safe speed will not be exceeded?

Prabhu Ramji

Numerade Educator

Problem 71

Your friend (mass $65.0 \mathrm{~kg}$ ) is standing on the ice in the middle of a frozen pond. There is very little friction between her feet and the ice, so she is unable to walk. Fortunately, a light rope is tied around her waist and you stand on the bank holding the other end. You pull on the rope for $3.00 \mathrm{~s}$ and accelerate your friend from rest to a speed of $6.00 \mathrm{~m} / \mathrm{s}$ while you remain at rest. What is the average power supplied by the force you applied?

Prabhu Ramji

Numerade Educator

Problem 72

On an essentially frictionless horizontal ice-skating rink, a skater moving at $3.0 \mathrm{~m} / \mathrm{s}$ encounters a rough patch that reduces her speed by $45 \%$ due to a friction force that is $25 \%$ of her weight. Use the work-energy theorem to find the length of the rough patch.

Prabhu Ramji

Numerade Educator

Problem 73

Pendulum. A small $0.12 \mathrm{~kg}$ metal ball is tied to a very light (essentially massless) string that is $0.8 \mathrm{~m}$ long. The string is attached to the ceiling so as to form a pendulum. The pendulum is set into motion by releasing it from rest at an angle of $60^{\circ}$ with the vertical. (a) What is the speed of the ball when it reaches the bottom of the arc? (b) What is the centripetal acceleration of the ball at this point? (c) What is the tension in the string at this point?

Prabhu Ramji

Numerade Educator

Problem 74

A pump is required to lift 750 liters of water per minute from a well $14.0 \mathrm{~m}$ deep and eject it with a speed of $18.0 \mathrm{~m} / \mathrm{s}$. How much work per minute does the pump do?

Abhishek Jana

Numerade Educator

Problem 75

A $350 \mathrm{~kg}$ roller coaster starts from rest at point $A$ and slides down the frictionless loop-the-loop shown in Figure 7.48 . (a) How fast is this roller coaster moving at point $B ?$ (b) How hard does it press against the track at point $B ?$

Andrew Puyleart

Andrew Puyleart

Numerade Educator

Problem 76

In action movies there are often chase scenes in which a car becomes airborne. When the car lands, its four suspension springs, one on each wheel, are compressed by the impact. For a typical passenger car, the suspension springs each have a spring constant of about $500 \mathrm{lb} / \mathrm{in} .$ and a maximum compression of $6 \mathrm{in} .$ Using this information, estimate the maximum height from which a $3300 \mathrm{lb}$ car could be dropped without the suspension springs exceeding their maximum compression. Assume that the mass of the car is distributed evenly among the four suspension springs.

Prabhu Ramji

Numerade Educator

Problem 77

In creating his definition of horsepower, James Watt, the inventor of the steam engine, calculated the power output of a horse operating a mill to grind grain or cut wood. The horse walked in a 24-ft-diameter circle, making, according to Watt, 144 trips around the circle in an hour. (a) Using the currently accepted value of 746 watts for 1 horsepower, calculate the force (in pounds) with which Mr. Watt's horse must have been pulling. (See Appendix D for useful conversion factors. $70 \mathrm{~kg}$ human being who climbs a 3.0 -m-high set of stairs in $5.0 \mathrm{~s}$

Prabhu Ramji

Numerade Educator

Problem 77

In creating his definition of horsepower, James Watt, the inventor of the steam engine, calculated the power output of a horse operating a mill to grind grain or cut wood. The horse walked in a 24-ft-diameter circle, making, according to Watt, 144 trips around the circle in an hour. (a) Using the currently accepted value of 746 watts for 1 horsepower, calculate the force (in pounds) with which Mr. Watt's horse must have been pulling. (See Appendix D for useful conversion factors. $70 \mathrm{~kg}$ human being who climbs a 3.0 -m-high set of stairs in $5.0 \mathrm{~s}$

Prabhu Ramji

Numerade Educator

Problem 78

All birds, independent of their size, must maintain a power output of $10-25$ watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (Patagona gigas) has mass $70 \mathrm{~g}$ and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A $70 \mathrm{~kg}$ athlete can maintain a power output of $1.4 \mathrm{~kW}$ for no more than a few seconds; the steady power output of a typical athlete is only $500 \mathrm{~W}$ or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

Paul A.

California State Polytechnic University, Pomona

Problem 79

A $250 \mathrm{~g}$ object on a frictionless, horizontal lab table is pushed against a spring of force constant $35 \mathrm{~N} / \mathrm{cm}$ and then released. Just before the object is released, the spring is compressed $12.0 \mathrm{~cm} .$ How fast is the object moving when it has gained half of the spring's original stored energy?

Prabhu Ramji

Numerade Educator

Problem 80

A bungee cord is $30.0 \mathrm{~m}$ long and, when stretched a distance $x$, it exerts a restoring force of magnitude $k x$. Your father-in-law (mass $95.0 \mathrm{~kg}$ ) stands on a platform $45.0 \mathrm{~m}$ above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only $41.0 \mathrm{~m}$ before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of $380.0 \mathrm{~N}$. When you do this, what distance will the bungee cord that you should select have stretched?

Prabhu Ramji

Numerade Educator

Problem 81

Riding a loop-the-loop. A car in an amusement park ride travels without friction along the track shown in Figure 7.49 , starting from rest at point $A .$ If the loop the car is currently on has a radius of $20.0 \mathrm{~m},$ find the minimum height $h$ so that the car will not fall off the track at the top of the circular part of the loop.

Prabhu Ramji

Numerade Educator

Problem 82

A $2.0 \mathrm{~kg}$ piece of wood slides on the surface shown in Figure 7.50 . All parts of the surface are frictionless, except for a 30 -m-long rough segment at the bottom, where the coefficient of kinetic friction with the wood is 0.20 . The wood starts from rest $4.0 \mathrm{~m}$ above the bottom. (a) Where will the wood eventually come to rest? (b) How much work is done by friction by the time the wood stops?

Prabhu Ramji

Numerade Educator

Problem 83

A $68 \mathrm{~kg}$ skier approaches the foot of a hill with a speed of $15 \mathrm{~m} / \mathrm{s}$. The surface of this hill slopes up at $40.0^{\circ}$ above the horizontal and has coefficients of static and kinetic friction of 0.75 and $0.25,$ respectively, with the skis. (a) Use energy conservation to find the maximum height above the foot of the hill that the skier will reach.
(b) Will the skier remain at rest once she stops, or will she begin to slide down the hill? Prove your answer.

Prabhu Ramji

Numerade Educator

Problem 84

A $70 \mathrm{~kg}$ hur ergy at the rate of $80 \mathrm{~J} / \mathrm{s},$ on average, for just resting When the person is engaged in more strenuous activities, be much higher. (a) If the individual did nothing but res food calories per day would she or he have to eat to make used up? (b) In what forms is energy used when a person sleeping? In other words, what happens to those $80 \mathrm{~J} / \mathrm{s} ?$ kinds of energy, mechanical and otherwise, do our body have? $(\mathrm{c})$ If an average person rested and did other lowfor 16 hours (which consumes $80 \mathrm{~J} / \mathrm{s}$ ) and did light activi for 8 hours (which consumes $200 \mathrm{~J} / \mathrm{s}$ ), how many calori or he have to consume per day to make up for the energy II The aircraft carrier USS George Washington has mass 1. When its engines are developing their full power of 260 George Washington travels at its top speed of 35 knots $70 \%$ of the power output of the engines is applied to pusl

Prabhu Ramji

Numerade Educator

Problem 85

The aircraft carrier USS George Washington has mass $1.0 \times 10^{8} \mathrm{~kg}$. When its engines are developing their full power of 260,000 hp, the George Washington travels at its top speed of 35 knots $(65 \mathrm{~km} / \mathrm{h})$. If $70 \%$ of the power output of the engines is applied to pushing the ship through the water, what is the magnitude of the force of water resistance that opposes the carrier's motion at this speed?

Prabhu Ramji

Numerade Educator

Problem 86

A ball is thrown upward with an initial velocity of $15 \mathrm{~m} / \mathrm{s}$ at an angle of $60.0^{\circ}$ above the horizontal. Use energy conservation to find the ball's greatest height above the ground.

Abhishek Jana

Numerade Educator

Problem 87

Automotive power. A truck engine transmits $28.0 \mathrm{~kW}(37.5 \mathrm{hp})$ to the driving wheels when the truck is traveling at a constant velocity of magnitude $60.0 \mathrm{~km} / \mathrm{h}(37.7 \mathrm{mi} / \mathrm{h})$ on a level road. (a) What is the resisting force acting on the truck? (b) Assume that $65 \%$ of the resisting force is due to rolling friction and the remainder is due to air resistance. If the force of rolling friction is independent of speed, and the force of air resistance is proportional to the square of the speed, what power will drive the truck at $30.0 \mathrm{~km} / \mathrm{h}$ ? At $120.0 \mathrm{~km} / \mathrm{h} ?$ Give your answers in kilowatts and in horsepower.

PS

Pramila Shakya

Numerade Educator

Problem 88

BIO Energy of locomotion. On flat ground, a $70 \mathrm{~kg}$ person requires about $300 \mathrm{~W}$ of metabolic power to walk at a steady pace of $5.0 \mathrm{~km} / \mathrm{h}(1.4 \mathrm{~m} / \mathrm{s})$ Using the same metabolic power output, that person can bicycle over the same ground at $15 \mathrm{~km} / \mathrm{h}$.
Based on these data, how does the energy used in biking $1 \mathrm{~km}$ co pare with that used in walking $1 \mathrm{~km} ?$ Biking takes
A. $\frac{1}{3}$ of the energy of walking the same distance.
B. the same energy as walking the same distance.
C. 3 times the energy of walking the same distance.
D. 9 times the energy of walking the same distance.

Prabhu Ramji

Numerade Educator

Problem 89

On flat ground, a $70 \mathrm{~kg}$ person requires about $300 \mathrm{~W}$ of metabolic power to walk at a steady pace of $5.0 \mathrm{~km} / \mathrm{h}(1.4 \mathrm{~m} / \mathrm{s})$ Using the same metabolic power output, that person can bicycle over the same ground at $15 \mathrm{~km} / \mathrm{h}$.
A $70 \mathrm{~kg}$ person walks at a steady pace of $5.0 \mathrm{~km} / \mathrm{h}$ on a treadmill at a $5.0 \%$ grade (that is, the vertical distance covered is $5.0 \%$ of the horizontal distance covered). If we assume the metabolic power required is equal to that required for walking on a flat surface plus the rate of doing work for the vertical climb, how much power is required?
A. $300 \mathrm{~W}$
B. 315 W
C. $350 \mathrm{~W}$
D. $370 \mathrm{~W}$

Prabhu Ramji

Numerade Educator

Problem 90

How many times greater is the kinetic energy of the person when biking than when walking? Ignore the mass of the bike.
A. 1.7
B. 3
C. 6
D. 9

Kara Merfeld

Numerade Educator

Problem 91

During the calibration process, the cantilever is observed to deflect by $0.10 \mathrm{nm}$ when a force of $3.0 \mathrm{pN}$ is applied to it. What deflection of the cantilever corresponds to a force of $6.0 \mathrm{pN} ?$
A. $0.07 \mathrm{nm}$
C. $0.20 \mathrm{nm}$
B. $0.14 \mathrm{nm}$
D. $0.40 \mathrm{nm}$

Prabhu Ramji

Numerade Educator

Problem 92

A segment of DNA is put in place and stretched. Figure 7.52 shows a graph of the force exerted on the DNA as a function of the displacement of the stage. Based on this graph, which statement is the best interpretation of the DNA's behavior over this range of displacements? The DNA
A. does not follow Hooke's law because its force constant increases as the force on it increases.
B. follows Hooke's law and has a force constant of about $0.1 \mathrm{pN} / \mathrm{nm}$
C. follows Hooke's law and has a force constant of about $10 \mathrm{pN} / \mathrm{nm}$
D. does not follow Hooke's law because its force constant decreases as the force on it increases.

Stephen Place

University of California, Irvine

Problem 93

Based on Figure 7.52 , how much elastic potential energy is stored in the DNA when it is stretched $50 \mathrm{nm} ?$
A. $2.5 \times 10^{-19} \mathrm{~J}$
B. $1.2 \times 10^{-19} \mathrm{~J}$
C. $5.0 \times 10^{-12} \mathrm{~J}$
D. $2.5 \times 10^{-12} \mathrm{~J}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 94

The stage moves at a constant speed while stretching the DNA. Which of the graphs in Figure 7.53 best represents the power supplied to the stage versus time?

Prabhu Ramji

Numerade Educator