University Physics Volume 1

Samuel J. Ling, Jeff Sanny, William Moebs

Chapter 7

Work and Kinetic Energy - all with Video Answers

Educators

+ 13 more educators

Chapter Questions

Problem 1

Give an example of something we think of as work in everyday circumstances that is not work in the scientific sense. Is energy transferred or changed in form in your example? If so, explain how this is accomplished without doing work.

Jacob Paiste

Jacob Paiste

Numerade Educator

Problem 2

Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work.

Jacob Paiste

Numerade Educator

Problem 3

Describe a situation in which a force is exerted for a long time but does no work. Explain.

Jacob Paiste

Numerade Educator

Problem 4

A body moves in a circle at constant speed. Does the centripetal force that accelerates the body do any work? Explain.

Jacob Paiste

Numerade Educator

Problem 5

Suppose you throw a ball upward and catch it when it returns at the same height. How much work does the gravitational force do on the ball over its entire trip?

Jacob Paiste

Numerade Educator

Problem 6

Why is it more difficult to do sit-ups while on a slant board than on a horizontal surface? (See below.)

Jacob Paiste

Numerade Educator

Problem 7

As a young man, Tarzan climbed up a vine to reach his tree house. As he got older, he decided to build and use a staircase instead. since the work of the gravitational force $m g$ is path independent, what did the King of the Apes gain in using stairs?

Jacob Paiste

Numerade Educator

Problem 8

A particle of $m$ has a velocity of $\hat{\mathbf{i}}+v_{y} \hat{\mathbf{j}}+v_{z} \hat{\mathbf{k}}$ Is its kinetic energy given by $m\left(v_{x}^{2} \hat{\mathbf{i}}+v_{y}^{2} \hat{\mathbf{j}}+v_{z}^{2} \hat{\mathbf{k}}\right) / 2 ?$ If not, what is the correct expression?

Jacob Paiste

Numerade Educator

Problem 9

One particle has mass $m$ and a second particle has mass 2 $m$. The second particle is moving with speed $v$ and the first with speed 2 v. How do their kinetic energies compare?

Jacob Paiste

Numerade Educator

Problem 10

A person drops a pebble of mass $m_{1}$ from a height $h,$ and it hits the floor with kinetic energy $K .$ The person drops another pebble of mass $m_{2}$ from a height of $2 h$, and it hits the floor with the same kinetic energy $K$. How do the masses of the pebbles compare?

Cheryl Glor

Numerade Educator

Problem 11

The person shown below does work on the lawn mower. Under what conditions would the mower gain energy from the person pushing the mower? Under what conditions would it lose energy?

Jacob Paiste

Numerade Educator

Problem 12

Work done on a system puts energy into it. Work done by a system removes energy from it. Give an example for each statement.

Jacob Paiste

Numerade Educator

Problem 13

Two marbles of masses $m$ and $2 \mathrm{m}$ are dropped from a height $h$. Compare their kinetic energies when they reach the ground.

Jacob Paiste

Numerade Educator

Problem 14

Compare the work required to accelerate a car of mass $2000 \mathrm{kg}$ from 30.0 to $40.0 \mathrm{km} / \mathrm{h}$ with that required for an acceleration from 50.0 to $60.0 \mathrm{km} / \mathrm{h}$.

Jacob Paiste

Numerade Educator

Problem 15

Suppose you are jogging at constant velocity. Are you doing any work on the environment and vice versa?

Jacob Paiste

Numerade Educator

Problem 16

Two forces act to double the speed of a particle, initially moving with kinetic energy of 1 J. One of the forces does $4 \mathrm{J}$ of work. How much work does the other force do?

Shoukat Ali

Other Schools

Problem 17

Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zero-watt device.) Explain in terms of the definition of power.

Jacob Paiste

Numerade Educator

Problem 18

Explain, in terms of the definition of power, why energy consumption is sometimes listed in kilowatt-hours rather than joules. What is the relationship between these two energy units?

Jacob Paiste

Numerade Educator

Problem 19

A spark of static electricity, such as that you might receive from a doorknob on a cold dry day, may carry a few hundred watts of power. Explain why you are not injured by such a spark.

Jacob Paiste

Numerade Educator

Problem 20

Does the work done in lifting an object depend on how fast it is lifted? Does the power expended depend on how fast it is lifted?

Jacob Paiste

Numerade Educator

Problem 21

Can the power expended by a force be negative?

Jacob Paiste

Numerade Educator

Problem 22

How can a 50-W light bulb use more energy than a 1000-W oven?

Jacob Paiste

Numerade Educator

Problem 23

How much work does a supermarket checkout attendant do on a can of soup he pushes $0.600 \mathrm{m}$ horizontally with a force of $5.00 \mathrm{N}$ ?

Jacob Paiste

Numerade Educator

Problem 24

A 75.0 -kg person climbs stairs, gaining $2.50 \mathrm{m}$ in height. Find the work done to accomplish this task.

Jacob Paiste

Numerade Educator

Problem 25

(a) Calculate the work done on a 1500 -kg elevator car by its cable to lift it $40.0 \mathrm{m}$ at constant speed, assuming friction averages $100 \mathrm{N}$. (b) What is the work done on the lift by the gravitational force in this process? (c) What is the total work done on the lift?

Mukesh Devi

Numerade Educator

Problem 26

Suppose a car travels $108 \mathrm{km}$ at a speed of $30.0 \mathrm{m} / \mathrm{s}$, and uses 2.0 gal of gasoline. Only $30 \%$ of the gasoline goes into useful work by the force that keeps the car moving at constant speed despite friction. (The energy content of gasoline is about 140 MJ/gal.) (a) What is the magnitude of the force exerted to keep the car moving at constant speed? (b) If the required force is directly proportional to speed, how many gallons will be used to drive $108 \mathrm{km}$ at a speed of $28.0 \mathrm{m} / \mathrm{s} ?$

Jacob Paiste

Numerade Educator

Problem 27

Calculate the work done by an 85.0 -kg man who pushes a crate $4.00 \mathrm{m}$ up along a ramp that makes an angle of $20.0^{\circ}$ with the horizontal (see below). He exerts a force of $500 \mathrm{N}$ on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate and on his body to get up the ramp.

Jacob Paiste

Numerade Educator

Problem 28

How much work is done by the boy pulling his sister $30.0 \mathrm{m}$ in a wagon as shown below? Assume no friction acts on the wagon.

Jacob Paiste

Numerade Educator

Problem 29

A shopper pushes a grocery cart 20.0 $\mathrm{m}$ at constant speed on level ground, against a 35.0 N frictional force. He pushes in a direction $25.0^{\circ}$ below the horizontal. (a) What is the work done on the cart by friction? (b) What is the work done on the cart by the gravitational force? (c) What is the work done on the cart by the shopper? (d) Find the force the shopper exerts, using energy considerations. (e) What is the total work done on the cart?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 30

Suppose the ski patrol lowers a rescue sled and victim, having a total mass of $90.0 \mathrm{kg}$, down a $60.0^{\circ}$ slope at constant speed, as shown below. The coefficient of friction between the sled and the snow is 0.100 . (a) How much work is done by friction as the sled moves $30.0 \mathrm{m}$ along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done?

Jacob Paiste

Numerade Educator

Problem 31

A constant 20-N force pushes a small ball in the direction of the force over a distance of $5.0 \mathrm{m}$. What is the work done by the force?

Jacob Paiste

Numerade Educator

Problem 32

A toy cart is pulled a distance of $6.0 \mathrm{m}$ in a straight line across the floor. The force pulling the cart has a magnitude of $20 \mathrm{N}$ and is directed at $37^{\circ}$ above the horizontal. What is the work done by this force?

Jacob Paiste

Numerade Educator

Problem 33

A 5.0-kg box rests on a horizontal surface. The coefficient of kinetic friction between the box and surface is $\mu_{K}=0.50 .$ A horizontal force pulls the box at constant velocity for $10 \mathrm{cm} .$ Find the work done by (a) the applied horizontal force, (b) the frictional force, and (c) the net force.

Jacob Paiste

Numerade Educator

Problem 34

A sled plus passenger with total mass $50 \mathrm{kg}$ is pulled $20 \mathrm{m}$ across the snow $\left(\mu_{k}=0.20\right)$ at constant velocity by a force directed $25^{\circ}$ above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Luis Rios

Numerade Educator

Problem 35

Suppose that the sled plus passenger of the preceding problem is pushed $20 \mathrm{m}$ across the snow at constant velocity by a force directed $30^{\circ}$ below the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Luis Rios

Numerade Educator

Problem 36

How much work does the force $F(x)=(-2.0 / x) \mathrm{N}$ do on a particle as it moves from $x=2.0 \mathrm{m}$ to $x=5.0 \mathrm{m} ?$

Jacob Paiste

Numerade Educator

Problem 37

How much work is done against the gravitational force on a 5.0 -kg briefcase when it is carried from the ground floor to the roof of the Empire State Building, a vertical climb of $380 \mathrm{m} ?$

Jacob Paiste

Numerade Educator

Problem 38

It takes $500 \mathrm{J}$ of work to compress a spring $10 \mathrm{cm}$. What is the force constant of the spring?

Eduard Sanchez

Numerade Educator

Problem 39

A bungee cord is essentially a very long rubber band that can stretch up to four times its unstretched length. However, its spring constant varies over its stretch [see Menz, P.G. "The Physics of Bungee Jumping." The Physics Teacher (November 1993) 31: 483-487]. Take the length of the cord to be along the $x$ -direction and define the stretch $x$ as the length of the cord $l$ minus its un-stretched length $l_{0} ;$ that is, $x=l-l_{0}$ (see below). Suppose a particular bungee cord has a spring constant, for $0 \leq x \leq 4.88 \mathrm{m},$ of $k_{1}=204 \mathrm{N} / \mathrm{m}$ and for $4.88 \mathrm{m} \leq x,$ of $k_{2}=111 \mathrm{N} / \mathrm{m}$
(Recall that the spring constant is the slope of the force $F(x) \text { versus its stretch } x .)$ (a) What is the tension in the cord when the stretch is $16.7 \mathrm{m}$ (the maximum desired for a given jump)? (b) How much work must be done against the elastic force of the bungee cord to stretch it $16.7 \mathrm{m} ?$

Jacob Paiste

Numerade Educator

Problem 40

A bungee cord exerts a nonlinear elastic force of magnitude $F(x)=k_{1} x+k_{2} x^{3}, \quad$ where $x$ is the distance the cord is stretched, $k_{1}=204 \mathrm{N} / \mathrm{m} \quad$ and $k_{2}=-0.233 \mathrm{N} / \mathrm{m}^{3} .$ How much work must be done on the cord to stretch it $16.7 \mathrm{m} ?$

Jacob Paiste

Numerade Educator

Problem 41

Engineers desire to model the magnitude of the elastic force of a bungee cord using the equation
$$
F(x)=a\left[\frac{x+9 m}{9 m}-\left(\frac{9 m}{x+9 m}\right)^{2}\right]
$$
where $x$ is the stretch of the cord along its length and $a$ is a constant. If it takes $22.0 \mathrm{kJ}$ of work to stretch the cord by 16.7 m, determine the value of the constant $a$

Supratim Pal

Numerade Educator

Problem 42

A particle moving in the $x y$ -plane is subject to a force
$$
\overrightarrow{\mathbf{F}}(x, y)=\left(50 \mathrm{N} \cdot \mathrm{m}^{2}\right) \frac{(x \hat{\mathbf{i}}+y \hat{\mathbf{j}})}{\left(x^{2}+y^{2}\right)^{3 / 2}}
$$
where $x$ and $y$ are in meters. Calculate the work done on the particle by this force, as it moves in a straight line from the point $(3 \mathrm{m}, 4 \mathrm{m})$ to the point $(8 \mathrm{m}, 6 \mathrm{m})$

Supratim Pal

Numerade Educator

Problem 43

A particle moves along a curved path $y(x)=(10 \mathrm{m})\left\{1+\cos \left[\left(0.1 \mathrm{m}^{-1}\right) x\right]\right\}, \quad$ from $\quad x=0$ $x=10 \pi \mathrm{m},$ subject to a tangential force of variable magnitude $\quad F(x)=(10 \mathrm{N}) \sin \left[\left(0.1 \mathrm{m}^{-1}\right) x\right] . \quad$ How much
work does the force do? (Hint: Consult a table of integrals or use a numerical integration program.)

Lucas Finney

Numerade Educator

Problem 44

Compare the kinetic energy of a 20,000 -kg truck moving at $110 \mathrm{km} / \mathrm{h}$ with that of an $80.0-\mathrm{kg}$ astronaut in orbit moving at $27,500 \mathrm{km} / \mathrm{h}$.

Jacob Adamczyk

Numerade Educator

Problem 45

(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0 -kg sprinter running at $10.0 \mathrm{m} / \mathrm{s}$ ? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 46

Estimate the kinetic energy of a 90,000 -ton aircraft carrier moving at a speed of at 30 knots. You will need to look up the definition of a nautical mile to use in converting the unit for speed, where 1 knot equals 1 nautical mile per hour.

Jacob Paiste

Numerade Educator

Problem 47

Calculate the kinetic energies of (a) a 2000.0-kg automobile moving at $100.0 \mathrm{km} / \mathrm{h} ;$ (b) an $80 .$ -kg runner sprinting at $10 . \mathrm{m} / \mathrm{s} ;$ and $(\mathrm{c})$ a $9.1 \times 10^{-31}$ -kg electron moving at $2.0 \times 10^{7} \mathrm{m} / \mathrm{s}$

Jacob Paiste

Numerade Educator

Problem 48

A 5.0-kg body has three times the kinetic energy of an 8.0 -kg body. Calculate the ratio of the speeds of these bodies.

Jacob Paiste

Numerade Educator

Problem 49

An 8.0-g bullet has a speed of 800 m/s. (a) What is its kinetic energy? (b) What is its kinetic energy if the speed is halved?

Jacob Paiste

Numerade Educator

Problem 50

(a) Calculate the force needed to bring a 950-kg car to rest from a speed of $90.0 \mathrm{km} / \mathrm{h}$ in a distance of $120 \mathrm{m}$ (a fairly typical distance for a non-panic stop). (b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in $2.00 \mathrm{m}$. Calculate the force exerted on the car and compare it with the force found in part (a).

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 51

A car's bumper is designed to withstand a 4.0-km/ h (1.1-m/s) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses $0.200 \mathrm{m}$ while bringing a $900-\mathrm{kg}$ car to rest from an initial speed of $1.1 \mathrm{m} / \mathrm{s}$

Jacob Paiste

Numerade Educator

Problem 52

Boxing gloves are padded to lessen the force of a blow. (a) Calculate the force exerted by a boxing glove on an opponent's face, if the glove and face compress 7.50 cm during a blow in which the 7.00 -kg arm and glove are brought to rest from an initial speed of $10.0 \mathrm{m} / \mathrm{s}$. (b) Calculate the force exerted by an identical blow in the gory old days when no gloves were used, and the knuckles and face would compress only $2.00 \mathrm{cm} .$ Assume the change in mass by removing the glove is negligible. (c) Discuss the magnitude of the force with glove on. Does it seem high enough to cause damage even though it is lower than the force with no glove?

Supratim Pal

Numerade Educator

Problem 53

Using energy considerations, calculate the average force a 60.0 -kg sprinter exerts backward on the track to accelerate from 2.00 to $8.00 \mathrm{m} / \mathrm{s}$ in a distance of $25.0 \mathrm{m}$, if he encounters a headwind that exerts an average force of 30.0 N against him.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 54

A 5.0-kg box has an acceleration of $2.0 \mathrm{m} / \mathrm{s}^{2}$ when it is pulled by a horizontal force across a surface with $\mu_{K}=0.50 .$ Find the work done over a distance of $10 \mathrm{cm}$ by (a) the horizontal force, (b) the frictional force, and (c) the net force. (d) What is the change in kinetic energy of the box?

Luis Rios

Numerade Educator

Problem 55

A constant 10-N horizontal force is applied to a 20-kg cart at rest on a level floor. If friction is negligible, what is the speed of the cart when it has been pushed 8.0 $\mathrm{m} ?$

Jacob Paiste

Numerade Educator

Problem 56

In the preceding problem, the 10 -N force is applied at an angle of $45^{\circ}$ below the horizontal. What is the speed of the cart when it has been pushed $8.0 \mathrm{m}$ ?

Jacob Paiste

Numerade Educator

Problem 57

Compare the work required to stop a 100 -kg crate sliding at $1.0 \mathrm{m} / \mathrm{s}$ and an 8.0 -g bullet traveling at $500 \mathrm{m} / \mathrm{s}$.

Jacob Paiste

Numerade Educator

Problem 58

A wagon with its passenger sits at the top of a hill. The wagon is given a slight push and rolls $100 \mathrm{m}$ down a $10^{\circ}$ incline to the bottom of the hill. What is the wagon's speed when it reaches the end of the incline. Assume that the retarding force of friction is negligible.

Jacob Paiste

Numerade Educator

Problem 59

An 8.0-g bullet with a speed of 800 m/s is shot into a wooden block and penetrates $20 \mathrm{cm}$ before stopping. What is the average force of the wood on the bullet? Assume the block does not move.

Ankur S

Numerade Educator

Problem 60

A 2.0-kg block starts with a speed of $10 \mathrm{m} / \mathrm{s}$ at the bottom of a plane inclined at $37^{\circ}$ to the horizontal. The coefficient of sliding friction between the block and plane is $\mu_{k}=0.30 .$ (a) Use the work-energy principle to determine how far the block slides along the plane before momentarily coming to rest. (b) After stopping, the block slides back down the plane. What is its speed when it reaches the bottom? (Hint: For the round trip, only the force of friction does work on the block.)

Jacob Paiste

Numerade Educator

Problem 61

When a 3.0-kg block is pushed against a massless spring of force constant constant $4.5 \times 10^{3} \mathrm{N} / \mathrm{m},$ the spring is compressed $8.0 \mathrm{cm} .$ The block is released, and it slides $2.0 \mathrm{m}$ (from the point at which it is released) across a horizontal surface before friction stops it. What is the coefficient of kinetic friction between the block and the surface?

Jacob Paiste

Numerade Educator

Problem 62

A small block of mass 200 g starts at rest at A, slides to $\mathrm{B}$ where its speed is $v_{B}=8.0 \mathrm{m} / \mathrm{s},$ then slides along the horizontal surface a distance $10 \mathrm{m}$ before coming to rest at C. (See below.) (a) What is the work of friction along the curved surface? (b) What is the coefficient of kinetic along the horizontal surface?

Jacob Paiste

Numerade Educator

Problem 63

A small object is placed at the top of an incline that is essentially frictionless. The object slides down the incline onto a rough horizontal surface, where it stops in $5.0 \mathrm{s}$ after traveling 60 $\mathrm{m}$. (a) What is the speed of the object at the bottom of the incline and its acceleration along the horizontal surface? (b) What is the height of the incline?

Jacob Paiste

Numerade Educator

Problem 64

When released, a $100-\mathrm{g}$ block slides down the path shown below, reaching the bottom with a speed of $4.0 \mathrm{m} / \mathrm{s}$. How much work does the force of friction do?

Ankur S

Numerade Educator

Problem 65

A 0.22LR-caliber bullet like that mentioned in Example 7.10 is fired into a door made of a single thickness of 1-inch pine boards. How fast would the bullet be traveling after it penetrated through the door?

Jacob Paiste

Numerade Educator

Problem 66

A sled starts from rest at the top of a snow-covered incline that makes a $22^{\circ}$ angle with the horizontal. After sliding $75 \mathrm{m}$ down the slope, its speed is $14 \mathrm{m} / \mathrm{s}$. Use the work-energy theorem to calculate the coefficient of kinetic friction between the runners of the sled and the snowy surface.

Supratim Pal

Numerade Educator

Problem 67

A person in good physical condition can put out 100 W of useful power for several hours at a stretch, perhaps by pedaling a mechanism that drives an electric generator. Neglecting any problems of generator efficiency and practical considerations such as resting time: (a) How many people would it take to run a 4.00 -kW electric clothes dryer? (b) How many people would it take to replace a large electric power plant that generates $800 \mathrm{MW}$ ?

Jacob Adamczyk

Numerade Educator

Problem 68

What is the cost of operating a $3.00-\mathrm{W}$ electric clock for a year if the cost of electricity is $\$ 0.0900$ per $\mathrm{kW} \cdot \mathrm{h}$ ?

Jacob Adamczyk

Numerade Educator

Problem 69

A large household air conditioner may consume 15.0 kW of power. What is the cost of operating this air conditioner 3.00 h per day for 30.0 d if the cost of electricity is $\$ 0.110$ per $k W \cdot h ?$

Jacob Paiste

Numerade Educator

Problem 70

(a) What is the average power consumption in watts of an appliance that uses $5.00 \mathrm{kW} \cdot \mathrm{h}$ of energy per day? $(\mathrm{b})$ How many joules of energy does this appliance consume in
a year?

Jacob Paiste

Numerade Educator

Problem 71

(a) What is the average useful power output of a person who does $6.00 \times 10^{6} \mathrm{J}$ of useful work in $8.00 \mathrm{h}$ ?
(b) Working at this rate, how long will it take this person to lift $2000 \mathrm{kg}$ of bricks $1.50 \mathrm{m}$ to a platform? (Work done to lift his body can be omitted because it is not considered useful output here.)

Jacob Paiste

Numerade Educator

Problem 72

A 500-kg dragster accelerates from rest to a final speed of $110 \mathrm{m} / \mathrm{s}$ in $400 \mathrm{m}$ (about a quarter of a mile) and encounters an average frictional force of 1200 N. What is its average power output in watts and horsepower if this takes $7.30 \mathrm{s} ?$

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 73

(a) How long will it take an 850-kg car with a useful power output of $40.0 \mathrm{hp}$ (1 hp equals $746 \mathrm{W}$ ) to reach a speed of $15.0 \mathrm{m} / \mathrm{s},$ neglecting friction? (b) How long will this acceleration take if the car also climbs a 3.00 -m high hill in the process?

Jacob Paiste

Numerade Educator

Problem 74

(a) Find the useful power output of an elevator motor that lifts a 2500 -kg load a height of $35.0 \mathrm{m}$ in $12.0 \mathrm{s}$, if it also increases the speed from rest to $4.00 \mathrm{m} / \mathrm{s}$. Note that the total mass of the counterbalanced system is $10,000 \mathrm{kg}-\mathrm{so}$ that only $2500 \mathrm{kg}$ is raised in height, but the full $10,000 \mathrm{kg}$ is accelerated. (b) What does it cost, if electricity is $\$ 0.0900$ per $\mathrm{kW} \cdot \mathrm{h}$ ?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 75

(a) How long would it take a $1.50 \times 10^{5}$ -kg airplane with engines that produce $100 \mathrm{MW}$ of power to reach a speed of $250 \mathrm{m} / \mathrm{s}$ and an altitude of $12.0 \mathrm{km}$ if air resistance were negligible? (b) If it actually takes 900 s, what is the power? (c) Given this power, what is the average force of air resistance if the airplane takes 1200 s? (Hint: You must find the distance the plane travels in 1200 s assuming constant acceleration.)

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 76

Calculate the power output needed for a $950-\mathrm{kg}$ car to climb a $2.00^{\circ}$ slope at a constant $30.0 \mathrm{m} / \mathrm{s}$ while encountering wind resistance and friction totaling $600 \mathrm{N}$

Jacob Paiste

Numerade Educator

Problem 77

A man of mass $80 \mathrm{kg}$ runs up a flight of stairs $20 \mathrm{m}$ high in 10 s. (a) how much power is used to lift the man? $(b)$ If the man's body is $25 \%$ efficient, how much power does he expend?

Jacob Paiste

Numerade Educator

Problem 78

The man of the preceding problem consumes approximately $\quad 1.05 \times 10^{7} \mathrm{J} \quad$ (2500 food calories) of energy per day in maintaining a constant weight. What is the average power he produces over a day? Compare this with his power production when he runs up the stairs.

Jacob Paiste

Numerade Educator

Problem 79

An electron in a television tube is accelerated uniformly from rest to a speed of $8.4 \times 10^{7} \mathrm{m} / \mathrm{s}$ over a distance of $2.5 \mathrm{cm} .$ What is the power delivered to the electron at the instant that its displacement is $1.0 \mathrm{cm} ?$

Luis Rios

Numerade Educator

Problem 80

Coal is lifted out of a mine a vertical distance of $50 \mathrm{m}$ by an engine that supplies $500 \mathrm{W}$ to a conveyer belt. How much coal per minute can be brought to the surface? Ignore the effects of friction.

Jacob Paiste

Numerade Educator

Problem 81

A girl pulls her 15-kg wagon along a flat sidewalk by applying a $10-\mathrm{N}$ force at $37^{\circ}$ to the horizontal. Assume that friction is negligible and that the wagon starts from rest. (a) How much work does the girl do on the wagon in the first $2.0 \mathrm{s}$. (b) How much instantaneous power does she exert at $t=2.0 \mathrm{s} ?$

Vishal Gupta

Numerade Educator

Problem 82

A typical automobile engine has an efficiency of 25\%. Suppose that the engine of a 1000 -kg automobile has a maximum power output of 140 hp. What is the maximum grade that the automobile can climb at $50 \mathrm{km} / \mathrm{h}$ if the frictional retarding force on it is $300 \mathrm{N}$ ?

Dading Chen

Numerade Educator

Problem 83

When jogging at $13 \mathrm{km} / \mathrm{h}$ on a level surface, a $70-\mathrm{kg}$ man uses energy at a rate of approximately 850 W. Using the facts that the "human engine" is approximately $25 \%$ efficient, determine the rate at which this man uses energy when jogging up a $5.0^{\circ}$ slope at this same speed. Assume that the frictional retarding force is the same in both cases.

Narayan Hari

Numerade Educator

Problem 84

A cart is pulled a distance $D$ on a flat, horizontal surface by a constant force $F$ that acts at an angle $\theta$ with the horizontal direction. The other forces on the object during this time are gravity $\left(F_{w}\right),$ normal forces $\left(F_{N 1}\right)$ and $\left(F_{N 2}\right),$ and rolling frictions $F_{r 1}$ and $F_{r 2},$ as shown below. What is the work done by each force?

Supratim Pal

Numerade Educator

Problem 85

Consider a particle on which several forces act, one of which is known to be constant in time:
$\overrightarrow{\mathbf{F}}_{1}=(3 \mathrm{N}) \hat{\mathbf{i}}+(4 \mathrm{N}) \hat{\mathbf{j}} .$ As a result, the particle moves along the $x$ -axis from $x=0$ to $x=5 \mathrm{m}$ in some time interval. What is the work done by $\overrightarrow{\mathbf{F}}_{1}$ ?

Jacob Paiste

Numerade Educator

Problem 86

Consider a particle on which several forces act, one of which is known to be constant in time:
$\overrightarrow{\mathbf{F}}_{1}=(3 \mathrm{N}) \hat{\mathbf{i}}+(4 \mathrm{N}) \hat{\mathbf{j}} .$ As a result, the particle moves first along the $x$ -axis from $x=0$ to $x=5 \mathrm{m}$ and then parallel to the $y$ -axis from $y=0$ to $y=6 \mathrm{m}$. What is the work done by $\overrightarrow{\mathbf{F}}_{1} ?$

Vishal Gupta

Numerade Educator

Problem 87

Consider a particle on which several forces act, one of which is known to be constant in time:
$\overrightarrow{\mathbf{F}}_{1}=(3 \mathrm{N}) \hat{\mathbf{i}}+(4 \mathrm{N}) \hat{\mathbf{j}} .$ As a result, the particle moves along a straight path from a Cartesian coordinate of (0 $\mathrm{m}, 0$
$\mathrm{m}$ ) to $(5 \mathrm{m}, 6 \mathrm{m}) .$ What is the work done by $\overrightarrow{\mathbf{F}}_{1} ?$

Jacob Paiste

Numerade Educator

Problem 88

Consider a particle on which a force acts that depends on the position of the particle. This force is given by $\overrightarrow{\mathbf{F}}_{1}=(2 y) \hat{\mathbf{i}}+(3 x) \hat{\mathbf{j}} .$ Find the work done by this force
when the particle moves from the origin to a point 5 meters to the right on the $x$ -axis.

Jacob Paiste

Numerade Educator

Problem 89

A boy pulls a 5-kg cart with a 20-N force at an angle of $30^{\circ}$ above the horizontal for a length of time. Over this time frame, the cart moves a distance of $12 \mathrm{m}$ on the horizontal floor. (a) Find the work done on the cart by the boy. (b) What will be the work done by the boy if he pulled with the same force horizontally instead of at an angle of $30^{\circ}$ above the horizontal over the same distance?

Jacob Paiste

Numerade Educator

Problem 90

A crate of mass $200 \mathrm{kg}$ is to be brought from a site on the ground floor to a third floor apartment. The workers know that they can either use the elevator first, then slide it along the third floor to the apartment, or first slide the crate to another location marked $\mathrm{C}$ below, and then take the elevator to the third floor and slide it on the third floor a shorter distance. The trouble is that the third floor is very rough compared to the ground floor. Given that the coefficient of kinetic friction between the crate and the ground floor is 0.100 and between the crate and the third floor surface is $0.300,$ find the work needed by the workers for each path shown from $A$ to $E$. Assume that the force the workers need to do is just enough to slide the crate at constant velocity (zero acceleration). Note: The work by the elevator against the force of gravity is not done by the workers.

Jacob Paiste

Numerade Educator

Problem 91

A hockey puck of mass 0.17 kg is shot across a rough floor with the roughness different at different places, which can be described by a position-dependent coefficient of kinetic friction. For a puck moving along the $x$ -axis, the coefficient of kinetic friction is the following function of $x$ where $x$ is in $\mathrm{m}: \mu(x)=0.1+0.05 x .$ Find the work done by the kinetic frictional force on the hockey puck when it has moved (a) from $x=0$ to $x=2 \mathrm{m},$ and (b) from $x=2 \mathrm{m}$ to $x=4 \mathrm{m}$

Jacob Paiste

Numerade Educator

Problem 92

A horizontal force of 20 $\mathrm{N}$ is required to keep a 5.0 kg box traveling at a constant speed up a frictionless incline for a vertical height change of $3.0 \mathrm{m}$. (a) What is the work done by gravity during this change in height? (b) What is the work done by the normal force? (c) What is the work done by the horizontal force?

Jacob Paiste

Numerade Educator

Problem 93

A 7.0-kg box slides along a horizontal frictionless floor at $1.7 \mathrm{m} / \mathrm{s}$ and collides with a relatively massless spring that compresses $23 \mathrm{cm}$ before the box comes to a stop. (a) How much kinetic energy does the box have before it collides with the spring? (b) Calculate the work done by the spring. (c) Determine the spring constant of the spring.

Supratim Pal

Numerade Educator

Problem 94

You are driving your car on a straight road with a coefficient of friction between the tires and the road of 0.55 A large piece of debris falls in front of your view and you immediate slam on the brakes, leaving a skid mark of 30.5 $\mathrm{m}$ (100-feet) long before coming to a stop. A policeman sees your car stopped on the road, looks at the skid mark, and gives you a ticket for traveling over the $13.4 \mathrm{m} / \mathrm{s}$ ( 30mph) speed limit. Should you fight the speeding ticket in court?

Jacob Paiste

Numerade Educator

Problem 95

A crate is being pushed across a rough floor surface. If no force is applied on the crate, the crate will slow down and come to a stop. If the crate of mass $50 \mathrm{kg}$ moving at speed $8 \mathrm{m} / \mathrm{s}$ comes to rest in 10 seconds, what is the rate at which the frictional force on the crate takes energy away from the crate?

Jacob Paiste

Numerade Educator

Problem 96

Suppose a horizontal force of $20 \mathrm{N}$ is required to maintain a speed of $8 \mathrm{m} / \mathrm{s}$ of a $50 \mathrm{kg}$ crate. (a) What is the power of this force? (b) Note that the acceleration of the crate is zero despite the fact that $20 \mathrm{N}$ force acts on the crate horizontally. What happens to the energy given to the crate as a result of the work done by this $20 \mathrm{N}$ force?

Jacob Paiste

Numerade Educator

Problem 97

Grains from a hopper falls at a rate of $10 \mathrm{kg} / \mathrm{s}$ vertically onto a conveyor belt that is moving horizontally at a constant speed of $2 \mathrm{m} / \mathrm{s}$. (a) What force is needed to keep the conveyor belt moving at the constant velocity? (b) What is the minimum power of the motor driving the conveyor belt?

Jacob Paiste

Numerade Educator

Problem 98

A cyclist in a race must climb a $5^{\circ}$ hill at a speed of 8 $\mathrm{m} / \mathrm{s}$. If the mass of the bike and the biker together is $80 \mathrm{kg}$. what must be the power output of the biker to achieve the goal?

Jacob Paiste

Numerade Educator

Problem 99

Shown below is a 40-kg crate that is pushed at constant velocity a distance $8.0 \mathrm{m}$ along a $30^{\circ}$ incline by the horizontal force $\overrightarrow{\mathbf{F}}$. The coefficient of kinetic friction between the crate and the incline is $\mu_{k}=0.40 .$ Calculate the work done by (a) the applied force, (b) the frictional force, (c) the gravitational force, and (d) the net force.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 100

The surface of the preceding problem is modified so that the coefficient of kinetic friction is decreased. The same horizontal force is applied to the crate, and after being pushed $8.0 \mathrm{m},$ its speed is $5.0 \mathrm{m} / \mathrm{s}$. How much work is now done by the force of friction? Assume that the crate starts at rest.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 101

The force $F(x)$ varies with position, as shown below. Find the work done by this force on a particle as it moves from $x=1.0 \mathrm{m}$ to $x=5.0 \mathrm{m}$

Nishant Kumar

Numerade Educator

Problem 102

Find the work done by the same force in Example \begin{tabular}{lllll} 7.4, & between & the & same & points, \\ \hline \end{tabular} $A=(0,0)$ and $B=(2 \mathrm{m}, 2 \mathrm{m}),$ over a circular arc of radius $2 \mathrm{m},$ centered at $(0,2 \mathrm{m}) .$ Evaluate the path integral using Cartesian coordinates. (Hint: You will probably need to consult a table of integrals.)

Vikash Ranjan

Numerade Educator

Problem 103

Answer the preceding problem using polar coordinates.

Sonam Khatri

Numerade Educator

Problem 104

Find the work done by the same force in Example \begin{tabular}{lllll} 7.4. & between & the & same & points, \\ \hline \end{tabular} $A=(0,0)$ and $B=(2 \mathrm{m}, 2 \mathrm{m}),$ over a circular arc of radius $2 \mathrm{m},$ centered at $(2 \mathrm{m}, 0) .$ Evaluate the path integral using Cartesian coordinates. (Hint: You will probably need to consult a table of integrals.)

Vikash Ranjan

Numerade Educator

Problem 105

Answer the preceding problem using polar the preceding problem using coordinates.

Paul Teng

Numerade Educator

Problem 106

Constant power $P$ is delivered to a car of mass $m$ by its engine. Show that if air resistance can be ignored, the distance covered in a time $t$ by the car, starting from rest, is given by $s=(8 P / 9 m)^{1 / 2} t^{3 / 2}$

Surjit Tewari

Numerade Educator

Problem 107

Suppose that the air resistance a car encounters is independent of its speed. When the car travels at $15 \mathrm{m} /$ s, its engine delivers 20 hp to its wheels. (a) What is the power delivered to the wheels when the car travels at $30 \mathrm{m} /$ s? (b) How much energy does the car use in covering 10 $\mathrm{km}$ at $15 \mathrm{m} / \mathrm{s}$ ? At $30 \mathrm{m} / \mathrm{s}$ ? Assume that the engine is $25 \%$ efficient. (c) Answer the same questions if the force of air resistance is proportional to the speed of the automobile. (d) What do these results, plus your experience with gasoline consumption, tell you about air resistance?

Prashant Bana

Numerade Educator

Problem 108

Consider a linear spring, as in Figure $7.7($ a), with mass $M$ uniformly distributed along its length. The left end of the spring is fixed, but the right end, at the equilibrium position $x=0, \quad$ is moving with speed $v$ in the $x$ -direction. What is the total kinetic energy of the spring? (Hint: First express the kinetic energy of an infinitesimal element of the spring $d m$ in terms of the total mass, equilibrium length, speed of the right-hand end, and position along the spring; then integrate.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator