Essential University Physics

Richard Wolfson

Chapter 6

Energy, Work, and Power - all with Video Answers

Educators

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Chapter Questions

Problem 1

If the scalar product of two nonzero vectors is zero, what can you conclude about their relative directions?

Donald Albin

Donald Albin

Numerade Educator

Problem 2

Must you do work to whirl a ball around on the end of a string? Explain.

Donald Albin

Numerade Educator

Problem 3

You want to raise a piano a given height using a ramp. With a fixed, nonzero coefficient of friction, will you have to do more work if the ramp is steeper or more gradual? Explain.

Donald Albin

Numerade Educator

Problem 4

Does the gravitational force of the Sun do work on a planet in a circular orbit? On a comet in an elliptical orbit? Explain.

Donald Albin

Numerade Educator

Problem 5

A pendulum bob swings back and forth on the end of a string, describing a circular arc. Does the tension force in the string do any work?

Donald Albin

Numerade Educator

Problem 6

Does your car's kinetic energy change if you drive at constant speed for 1 hour?

Donald Albin

Numerade Educator

Problem 7

A watt-second is a unit of what quantity? Relate it to a more standard SI unit.

Donald Albin

Numerade Educator

Problem 8

A truck is moving northward at $55 \mathrm{mi} / \mathrm{h}$. Later, it's moving eastward at the same speed. Has its kinetic energy changed? Has work been done on the truck? Has a force acted on the truck? Explain.

Donald Albin

Numerade Educator

Problem 9

A news article reports that a new solar farm will produce 143 kilowatt-hours of electricity. Criticize this statement. What did the writer probably mean?

Donald Albin

Numerade Educator

Problem 10

Is it possible for you to do work on an object without changing the object's kinetic energy? Explain.

Donald Albin

Numerade Educator

Problem 11

How much work do you do as you exert a 75-N force to push a shopping cart through a 12 -m-long supermarket aisle?

Donald Albin

Numerade Educator

Problem 12

If the coefficient of kinetic friction is $0.21,$ how much work do
you do when you slide a $50-\mathrm{kg}$ box at constant speed across a 4.8 -m-wide room?

Donald Albin

Numerade Educator

Problem 13

A crane lifts a 650 -kg beam vertically upward $23 \mathrm{~m}$ and then swings it eastward $18 \mathrm{~m}$. How much work does the crane do? Neglect friction, and assume the beam moves with constant speed.

Donald Albin

Numerade Educator

Problem 14

The world's highest waterfall, the Cherun-Meru in Venezuela, has a total drop of $980 \mathrm{~m} .$ How much work does gravity do on a cubic meter of water dropping down the Cherun-Meru?

Donald Albin

Numerade Educator

Problem 15

A meteorite plunges to Earth, embedding itself $75 \mathrm{~cm}$ in the ground. If it does $140 \mathrm{MJ}$ of work in the process, what average force does the meteorite exert on the ground?

Donald Albin

Numerade Educator

Problem 16

An elevator of mass $m$ rises a vertical distance $h$ with upward acceleration equal to one-tenth $g .$ Find an expression for the work the elevator cable does on the elevator.

Donald Albin

Numerade Educator

Problem 17

Show that the scalar product obeys the distributive law:
$\vec{A} \cdot(\vec{B}+\vec{C})=\vec{A} \cdot \vec{B}+\vec{A} \cdot \vec{C}$

Donald Albin

Numerade Educator

Problem 18

Find the work done by a force $\vec{F}=1.8 \hat{\imath}+2.2 \hat{\jmath} \mathrm{N}$ as it acts on an object moving from the origin to the point $56 \hat{\imath}+31 \hat{\jmath} \mathrm{m}$.

Nishant Kumar

Numerade Educator

Problem 19

To push a stalled car, you apply a $470-\mathrm{N}$ force at $17^{\circ}$ to the car's motion, doing $860 \mathrm{~J}$ of work in the process. How far do you push the car?

Donald Albin

Numerade Educator

Problem 20

Find the total work done by the force shown in Fig. 6.16 as the object on which it acts moves
(a) from $x=0$ to $x=3 \mathrm{~km}$ and
(b) from $x=3 \mathrm{~km}$ to $x=4 \mathrm{~km}$

Shoukat Ali

Other Schools

Problem 21

How much work does it take to stretch a spring with $k=200 \mathrm{~N} / \mathrm{m}($ a $) 10 \mathrm{~cm}$ from
equilibrium and (b) from $10 \mathrm{~cm}$ to $20 \mathrm{~cm}$ from equilibrium?

Donald Albin

Numerade Educator

Problem 22

Uncompressed, the spring for an automobile suspension is $45 \mathrm{~cm}$ long. It needs to be fitted into a space $32 \mathrm{~cm}$ long. If the spring constant is $3.8 \mathrm{kN} / \mathrm{m},$ how much work does a mechanic have to do to fit the spring?

Donald Albin

Numerade Educator

Problem 23

You do $8.5 \mathrm{~J}$ of work to stretch a spring with $k=190 \mathrm{~N} / \mathrm{m}$, starting with the spring unstretched. How far does the spring stretch?

Donald Albin

Numerade Educator

Problem 24

Spider silk is a remarkable elastic material. A particular strand has spring constant $70 \mathrm{mN} / \mathrm{m},$ and it stretches $9.6 \mathrm{~cm}$ when a fly hits it. How much work did the fly's impact do on the silk strand?

Donald Albin

Numerade Educator

Problem 25

What's the kinetic energy of a $2.4 \times 10^{5}$ -kg airplane cruising at $900 \mathrm{~km} / \mathrm{h} ?$

Donald Albin

Numerade Educator

Problem 26

A cyclotron accelerates protons from rest to $21 \mathrm{Mm} / \mathrm{s}$. How much work does it do on each proton?

Donald Albin

Numerade Educator

Problem 27

At what speed must a 950 -kg subcompact car be moving to have the same kinetic energy as a $3.2 \times 10^{4}-\mathrm{kg}$ truck going $20 \mathrm{~km} / \mathrm{h} ?$

Ankur S

Numerade Educator

Problem 28

A 60 -kg skateboarder comes over the top of a hill at $5.0 \mathrm{~m} / \mathrm{s}$ and reaches $10 \mathrm{~m} / \mathrm{s}$ at the bottom. Find the total work done on the
skateboarder between the top and bottom of the hill.

Donald Albin

Numerade Educator

Problem 29

After a tornado, a 0.50 -g drinking straw was found embedded 4.5 $\mathrm{cm}$ in a tree. Subsequent measurements showed that the tree exerted a stopping force of $70 \mathrm{~N}$ on the straw. What was the straw's speed?

Pawan Yadav

Numerade Educator

Problem 30

From what height would you have to drop a car for its impact to be equivalent to a $20-\mathrm{mi} / \mathrm{h}$ collision?

Donald Albin

Numerade Educator

Problem 31

A typical human diet is "2000 calories" per day, where the "calorie" describing food energy is actually 1 kilocalorie. Express 2000 kcal/day in watts.

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 32

A horse plows a 200 -m-long furrow in 5.0 min, exerting a $750-\mathrm{N}$ force. Find its power output, measured in watts and in horsepower.

Donald Albin

Numerade Educator

Problem 33

A typical car battery stores about $1 \mathrm{~kW} \cdot \mathrm{h}$ of energy. What's its power output if it drains completely in (a) 1 minute,
(b) 1 hour, and
(c) 1 day?

Shoukat Ali

Other Schools

Problem 34

A sprinter completes a 100 -m dash in 10.6 s, doing 22.4 kJ of work. What's her average power output?

Donald Albin

Numerade Educator

Problem 35

How much work can a 3.5 -hp lawnmower engine do in $1 \mathrm{~h}$ ?

Donald Albin

Numerade Educator

Problem 36

A 75 -kg long-jumper takes 3.1 s to reach a prejump speed of $10 \mathrm{~m} / \mathrm{s} .$ What's his power output?

Donald Albin

Numerade Educator

Problem 37

Estimate your power output as you do deep knee bends at the rate of one per second.

Donald Albin

Numerade Educator

Problem 38

In midday sunshine, solar energy strikes Earth at the rate of about $1 \mathrm{~kW} / \mathrm{m}^{2} .$ How long would it take a perfectly efficient solar collector of $15-\mathrm{m}^{2}$ area to collect $40 \mathrm{~kW} \cdot \mathrm{h}$ of energy? (Note: This is roughly the energy content of a gallon of gasoline.)

Donald Albin

Numerade Educator

Problem 39

It takes about $20 \mathrm{~kJ}$ to melt an ice cube. A typical microwave oven produces $900 \mathrm{~W}$ of microwave power. How long will it take a typical microwave to melt the ice cube?

Donald Albin

Numerade Educator

Problem 40

Which consumes more energy, a 1.2-kW hair dryer used for 10 min or a 7-W night-light left on for $24 \mathrm{~h} ?$

Vishal Gupta

Numerade Educator

Problem 41

A cord used in bungee jumping is normally $9.58 \mathrm{~m}$ long and has spring constant $k=235 \mathrm{~N} / \mathrm{m} .$ At the lowest point in a jump, the cord length has doubled. How much work has been done on the cord?

Shoukat Ali

Other Schools

Problem 42

Unstretched, a cord used in bungee jumping is $12.2 \mathrm{~m}$ long. When a jumper reaches the lowest point in her jump, the cord has stretched to $26.3 \mathrm{~m}$ and she's done $15.4 \mathrm{~kJ}$ of work on the cord. What's the spring constant of the cord?

Shoukat Ali

Other Schools

Problem 43

A 2.35- $\mu \mathrm{m}$ strand of DNA has an effective spring constant of $1.63 \times 10^{-7} \mathrm{~N} / \mathrm{m} .$ Find the work required to compress the strand so its length shrinks by $1.00 \%$.

Shoukat Ali

Other Schools

Problem 44

Find the effective spring constant of a DNA molecule, given that it takes $6.92 \times 10^{-24} \mathrm{~J}$ of work to compress it $4.48 \mathrm{nm}$.

Shoukat Ali

Other Schools

Problem 45

You and a partner are pedaling a 16.0 -kg tandem bicycle up a $6.22^{\circ}$ incline at $18.5 \mathrm{~km} / \mathrm{h} .$ There's a $10.8-\mathrm{N}$ force from air resistance. If the combined mass of you and your partner is $132 \mathrm{~kg}$, what power must the two of you supply?

Shoukat Ali

Other Schools

Problem 46

You and your cycling partner are capable of producing $955 \mathrm{~W}$ of power. What's the fastest you can pedal up a $4.40^{\circ}$ slope if the combined mass of your tandem bicycle and both riders is $152 \mathrm{~kg}$ and you face a $14.5-\mathrm{N}$ force from air resistance?

Shoukat Ali

Other Schools

Problem 47

A Boeing 787-9 jetliner has a mass of $245,000 \mathrm{~kg}$ including passengers. Its two engines produce a combined thrust force of $642 \mathrm{kN},$ and the aircraft cruises at $913 \mathrm{~km} / \mathrm{h}$ in level flight $-\mathrm{in}$ which case drag from the air is the only force the plane needs to overcome. Find the engines' power output (a) while cruising and (b) when it's climbing at a $23.0^{\circ}$ angle at $622 \mathrm{~km} / \mathrm{h}$. Assume air resistance doesn't change-although in reality it's greater at the higher speed.

Shoukat Ali

Other Schools

Problem 48

You're an aircraft designer charged with determining the maximum speed for a new aircraft when it's climbing at a $15.4^{\circ}$ angle. The total mass of the plane is $138,000 \mathrm{~kg},$ and its total engine power is $105 \mathrm{MW}$. While climbing it encounters a force of $193 \mathrm{kN}$ from air resistance. What do you report for the maximum speed while climbing?

Shoukat Ali

Other Schools

Problem 49

You slide a box of books at constant speed up a $30^{\circ}$ ramp, applying a force of $200 \mathrm{~N}$ directed up the slope. The coefficient of sliding friction is 0.18 . (a) How much work have you done when the box has risen $1 \mathrm{~m}$ vertically? (b) What's the mass of the box?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 50

Two people push a stalled car at its front doors, each applying a $280-\mathrm{N}$ force at $25^{\circ}$ to the forward direction, as shown in Fig. 6.17. How much work does each person do in pushing the car $5.6 \mathrm{~m} ?$

Donald Albin

Numerade Educator

Problem 51

You pull a box $23 \mathrm{~m}$ horizontally, using the rope shown in Fig. $6.18 .$ If the rope tension is $120 \mathrm{~N},$ and if the rope does $2500 \mathrm{~J}$ of work on the box, what angle $\theta$ does the rope make with the horizontal?

Donald Albin

Numerade Educator

Problem 52

(a) Find the scalar products $\hat{\imath} \cdot \hat{\imath}, \hat{\jmath} \cdot \hat{\jmath},$ and $\hat{k} \cdot \hat{k}$.
(b) Find $\hat{\imath} \cdot \hat{\jmath}, \hat{\jmath} \cdot \hat{k},$ and $\hat{k} \cdot \hat{\imath}$
(c) Use the distributive law to multiply out the scalar product of two arbitrary vectors $\vec{A}=A_{x} \hat{\imath}+A_{y} \hat{\jmath}+A_{z} \hat{k}$ and $\vec{B}=B_{x} \hat{\imath}+B_{y} \hat{\jmath}+B_{z} \hat{k}$
and use the results of (a) and (b) to verify Equation 6.4 .

Donald Albin

Numerade Educator

Problem 53

(a) Find the scalar product of the vectors $a \hat{\imath}+b \hat{\jmath}$ and $b \hat{\imath}-a \hat{\jmath}$ where $a$ and $b$ are arbitrary constants.
(b) What's the angle between the two vectors?

Donald Albin

Numerade Educator

Problem 54

A force $\vec{F}$ acts in the $x$ -direction, its magnitude given by $F=a x^{2}$ where $x$ is in meters and $a=5.0 \mathrm{~N} / \mathrm{m}^{2}$. Find the work done
by this force as it acts on a particle moving from $x=0$ to $x=6.0 \mathrm{~m}$

Donald Albin

Numerade Educator

Problem 55

A certain amount of work is required to stretch spring A a certain distance. Twice as much work is required to stretch spring B half that distance. Compare the spring constants of the two.

Donald Albin

Numerade Educator

Problem 56

A force with magnitude given by $F=a \sqrt{x}-b x^{2}$ acts in the $x$ -direction, where $a=25.2 \mathrm{~N} \cdot \mathrm{m}^{-1 / 2}$ and $b=3.87 \mathrm{~N} / \mathrm{m}^{2}$
Find the work this force does on an object moving from
(a) $x=0$ to $x=2.00 \mathrm{~m}$ and
(b) from $x=2.00 \mathrm{~m}$ to $x=3.75 \mathrm{~m}$

Shoukat Ali

Other Schools

Problem 57

The force exerted by a rubber band is given approximately by
$$
F=F_{0}\left[\frac{L_{0}-x}{L_{0}}-\frac{L_{0}^{2}}{\left(L_{0}+x\right)^{2}}\right]
$$
where $L_{0}$ is the unstretched length, $x$ is the stretch, and $F_{0}$ is a constant. Find an expression for the work needed to stretch the rubber band a distance $x$.

Donald Albin

Numerade Educator

Problem 58

You put your little sister (mass $m$ ) on a swing whose chains have length $L$ and pull slowly back until the swing makes an angle $\phi$ with the vertical. Show that the work you do is $m g L(1-\cos \phi)$

Donald Albin

Numerade Educator

Problem 59

Two unknown elementary particles pass through a detection chamber. If they have the same kinetic energy and their mass ratio is $4: 1,$ what's the ratio of their speeds?

Donald Albin

Numerade Educator

Problem 60

A tractor tows a plane from its airport gate, doing $8.7 \mathrm{MJ}$ of work. The link from the plane to the tractor makes a $22^{\circ}$ angle with the plane's motion, and the tension in the link is $0.41 \mathrm{MN}$. How far does the tractor move the plane?

Donald Albin

Numerade Educator

Problem 61

$E. coli$ bacteria swim by means of flagella that rotate about 100 times per second. A typical $E. coli$ bacterium swims at $22 \mu \mathrm{m} / \mathrm{s}$ its flagella exerting a force of $0.57 \mathrm{pN}$ to overcome the resistance due to its liquid environment. (a) What's the bacterium's power output? (b) How much work would it do in traversing the 25 -mm width of a microscope slide?

Donald Albin

Numerade Educator

Problem 62

On February $15,2013,$ an asteroid moving at $19 \mathrm{~km} / \mathrm{s}$ entered Earth's atmosphere over Chelyabinsk, Russia, and exploded at an altitude of more than $20 \mathrm{~km}$. This was the largest object known to have entered the atmosphere in over a century. The asteroid's kinetic energy just before entering the atmosphere was estimated as the energy equivalent of 500 kilotons of the explosive TNT. (Kilotons [kt] and megatons [Mt] are energy units used to describe the explosive yields of nuclear weapons, and you'll find the energy equivalent of 1 Mt in Appendix C.) What was the approximate mass of the Chelyabinsk asteroid?

Donald Albin

Numerade Educator

Problem 63

An elevator ascends from the ground floor to the 10 th floor, a height of $41 \mathrm{~m},$ in $35 \mathrm{~s}$. If the mass of the elevator and passengers is $840 \mathrm{~kg}$, what's the power necessary to lift the elevator? (Your answer is greater than the actual power needed because elevators are counterweighted, thus reducing the work the motor needs to do.)

Shoukat Ali

Other Schools

Problem 64

You're asked to assess the reliability of a nuclear power plant, as measured by the capacity factor-the ratio of the energy it actually produces to what it could produce if it operated all the time. The plant is rated at $840 \mathrm{MW}$ of electrical power output, and in a full year it produces $6.8 \times 10^{9} \mathrm{~kW} \cdot \mathrm{h}$ of electrical energy. What's its capacity factor?

Shoukat Ali

Other Schools

Problem 65

A force pointing in the $x$ -direction is given by $F=F_{0}\left(x / x_{0}\right)^{2}$ where $F_{0}$ and $x_{0}$ are constants and $x$ is position. Find an expression for the work done by this force as it acts on an object moving from $x=0$ to $x=x_{0}$

Donald Albin

Numerade Educator

Problem 66

A force pointing in the $x$ -direction is given by $F=a x^{3 / 2},$ where $a$ is a constant. The force does $1.86 \mathrm{~kJ}$ of work on an object as the object moves from $x=0$ to $x=18.5 \mathrm{~m} .$ Find the constant $a$.

Shoukat Ali

Other Schools

Problem 67

Two vectors have equal magnitude, and their scalar product is half the square of their magnitude. Find the angle between them.

Shoukat Ali

Other Schools

Problem 68

At what rate can a half-horsepower well pump deliver water to a tank $60 \mathrm{~m}$ above the water level in the well? Give your answer in $\mathrm{kg} / \mathrm{s}$ and $\mathrm{gal} / \mathrm{min} .$

Donald Albin

Numerade Educator

Problem 69

The United States imports about 400 million gallons of oil each day. Use the "Energy Content of Fuels" table in Appendix C to estimate the corresponding power, measured in gigawatts.

Shoukat Ali

Other Schools

Problem 70

By measuring oxygen uptake, sports physiologists have found that long-distance runners' power output is given approximately by $P=m(b v-c),$ where $m$ and $v$ are the runner's mass and speed, and $b$ and $c$ are constants given by $b=4.27 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{m}$ and $c=1.83 \mathrm{~W} / \mathrm{kg} .$ Determine the work done by a $54-\mathrm{kg}$ runner who runs a 10-km race at $5.2 \mathrm{~m} / \mathrm{s}$.

Donald Albin

Numerade Educator

Problem 71

The motor in a 1590 -kg Nissan Leaf electric car supplies energy to the wheels at the rate of $80.0 \mathrm{~kW}$. (a) What's the maximum speed for the Leaf when it's ascending an $11.8^{\circ}$ slope using only its electric motor? (b) What's the motor's power output in horsepower? Neglect air resistance.

Supratim Pal

Numerade Educator

Problem 72

A $1400-\mathrm{kg}$ car ascends a mountain road at a steady $60 \mathrm{~km} / \mathrm{h},$ against a $450-\mathrm{N}$ force of air resistance. If the engine supplies energy to the drive wheels at the rate of $38 \mathrm{~kW}$, what's the slope angle of the road?

Donald Albin

Numerade Educator

Problem 73

You do $2.2 \mathrm{~kJ}$ of work pushing a 78 -kg trunk at constant speed 3.1 $\mathrm{m}$ along a ramp inclined upward at $22^{\circ} .$ What's the frictional coefficient between trunk and ramp?

Donald Albin

Numerade Educator

Problem 74

(a) Find the work done in lifting 1 L of blood (mass 1 kg) from the foot to the head of a 1.7 -m-tall person. (b) If blood circulates through the body at the rate of $5.0 \mathrm{~L} / \mathrm{min},$ estimate the heart's power output. (Your answer underestimates the power by a factor of about 5 because it neglects fluid friction and other factors.)

Donald Albin

Numerade Educator

Problem 75

You push an 84.5 -kg chest of drawers at $0.386 \mathrm{~m} / \mathrm{s},$ for a distance of $6.55 \mathrm{~m}$ across a level floor, where the coefficient of friction is $0.612 .$ Find
(a) the power needed and
(b) the work you do.
(c) Repeat for the case of a ramp sloping upward at $5.75^{\circ},$ with all other quantities unchanged.

Shoukat Ali

Other Schools

Problem 76

You mix flour into bread dough, exerting a 45-N force on the spoon, which you move at $0.29 \mathrm{~m} / \mathrm{s}$. (a) What power do you supply?
(b) How much work do you do if you stir for $1.0 \mathrm{~min} ?$

Donald Albin

Numerade Educator

Problem 77

One machine does work at a constant rate $P_{0}$. A second machine does work at a rate given by $P(t)=2 P_{0}\left(1-\frac{\left(t-t_{0}\right)^{2}}{t_{0}^{2}}\right),$ where $t_{0}$ is a constant with the units of time. Both machines start at time $t=0 .$ Find expressions for (a) the peak power output of the second machine and (b) the earliest time at which both machines have done the same amount of work.

Shoukat Ali

Other Schools

Problem 78

A typical bumblebee has mass $0.25 \mathrm{mg}$. It beats its wings 100 times per second, and the wings undergo an average displacement of about $1.5 \mathrm{~mm}$. When the bee is hovering over a flower, the average force between wings and air must support the bee's weight. Estimate the average power the bee expends in hovering.

Donald Albin

Numerade Educator

Problem 79

You're trying to decide whether to buy an energy-efficient 225 -W refrigerator for $\$ 1150$ or a standard 425 -W model for $\$ 850 .$ The standard model will run $20 \%$ of the time, but better insulation means the energy-efficient model will run $11 \%$ of the time. If electricity costs $9.5 \mathrm{c} / \mathrm{kW} \cdot \mathrm{h}$ how long would you have to own the energy-efficient model to make up the difference in cost? Neglect interest you might earn on your money.

Brian Francisco

Brian Francisco

Numerade Educator

Problem 80

Your friend does five reps with a barbell, on each rep lifting $45 \mathrm{~kg} 0.50 \mathrm{~m} .$ She claims the work done is enough to "burn off" a chocolate bar with energy content 230 kcal (see Exercise 31 ). Is that true? If not, how many lifts would it take?

Shoukat Ali

Other Schools

Problem 81

A machine delivers power at a decreasing rate $P=P_{0} t_{0}^{2} /\left(t+t_{0}\right)^{2}$ where $P_{0}$ and $t_{0}$ are constants. The machine starts at $t=0$ and runs forever. Show that it nevertheless does only a finite amount of work, equal to $P_{0} t_{0}$.

Donald Albin

Numerade Educator

Problem 82

A locomotive accelerates a freight train of total mass $M$ from rest, applying constant power $P .$ Determine the speed and position of the train as functions of time, assuming all the power goes to increasing the train's kinetic energy.

Donald Albin

Numerade Educator

Problem 83

A force given by $F=b / \sqrt{x}$ acts in the $x$ -direction, where $b$ is a constant with the units $\mathrm{N} \cdot \mathrm{m}^{1 / 2}$. Show that even though the force becomes arbitrarily large as $x$ approaches zero, the work done in moving from $x_{1}$ to $x_{2}$ remains finite even as $x_{1}$ approaches zero. Find an expression for that work in the limit $x_{1} \rightarrow 0$.

Donald Albin

Numerade Educator

Problem 84

You're assisting a cardiologist in planning a stress test for a 75 kg patient. The test involves rapid walking on an inclined treadmill, and the patient is to reach a peak power output of $350 \mathrm{~W}$. If the patient's maximum walking speed is $8.0 \mathrm{~km} / \mathrm{h},$ what should be the treadmill's inclination angle?

Donald Albin

Numerade Educator

Problem 85

You're an engineer for a company that makes bungee-jump cords, and you're asked to develop a formula for the work involved in stretching cords to double their length. Your cords have forcedistance relations described by $F=-\left(k x+b x^{2}+c x^{3}+d x^{4}\right)$ where $k, b, c,$ and $d$ are constants.
(a) Given a cord with unstretched length $L_{0},$ what's your formula? (b) Evaluate the work done in doubling the stretch of a $10-\mathrm{m}$ cord with $k=420 \mathrm{~N} / \mathrm{m}$ $b=-86 \mathrm{~N} / \mathrm{m}^{2}, c=12 \mathrm{~N} / \mathrm{m}^{3},$ and $d=-0.50 \mathrm{~N} / \mathrm{m}^{4}$

Vishal Gupta

Numerade Educator

Problem 86

You push an object of mass $m$ slowly partway up a loop-the-loop track of radius $R$, starting from the bottom, and ending at a height $h<R$ above the bottom. The coefficient of friction between the object and the track is a constant. Show that the work you do against friction is $\mu m g \vee \sqrt{2 h R-h^{2}}$

Vishal Gupta

Numerade Educator

Problem 87

A particle moves from the origin to the point $x=3 \mathrm{~m}, y=6 \mathrm{~m}$ along the curve $y=a x^{2}-b x,$ where $a=2 \mathrm{~m}^{-1}$ and $b=4 .$ It's subject to a force $c x y \hat{\imath}+d \hat{\jmath},$ where $c=10 \mathrm{~N} / \mathrm{m}^{2}$ and $d=15 \mathrm{~N}$
Calculate the work done by the force.

Donald Albin

Numerade Educator

Problem 88

Repeat Problem 87 for the following cases: (a) the particle moves first along the $x$ -axis from the origin to the point $(3 \mathrm{~m}, 0)$ and then parallel to the $y$ -axis until it reaches $(3 \mathrm{~m}, 6 \mathrm{~m}) ;$ (b) it moves first along the $y$ -axis from the origin to the point $(0,6 \mathrm{~m})$ and then parallel to the $x$ -axis until it reaches $(3 \mathrm{~m}, 6 \mathrm{~m})$.

Shoukat Ali

Other Schools

Problem 89

The world's fastest elevator, in Taiwan's Taipei 101 skyscraper (Fig. 6.19 ), ascends at the rate of $1010 \mathrm{~m} / \mathrm{min}$. Counterweights balance the weight of the elevator car, so the motor doesn't have to lift the car's weight. If the motor produces $330 \mathrm{~kW}$ of power, what's the maximum number of 67 -kg people the elevator can accommodate? (Your answer somewhat overestimates the actual rated load of 24 people.)

Donald Albin

Numerade Educator

Problem 90

An experimental measurement of the force required to stretch a slingshot is given in the table below. Plot the force-distance curve for this slingshot and use graphical integration to determine the work done in stretching the slingshot the full $40-\mathrm{cm}$ distance.
$$
\begin{array}{cl}
\hline \text { Stretch (cm) } & \text { Force (N) } \\
\hline 0 & 0 \\
5.00 & 0.885 \\
10.0 & 1.89 \\
15.0 & 3.05 \\
20.0 & 4.48 \\
25.0 & 6.44 \\
30.0 & 8.22 \\
35.0 & 9.95 \\
40.0 & 12.7 \\
\hline
\end{array}
$$

Donald Albin

Numerade Educator

Problem 91

You're an expert witness in a medical malpractice lawsuit. A hospital patient's leg slipped off a stretcher, and his heel hit the floor. The defense attorney for the hospital claims the leg, with mass 8 $\mathrm{kg},$ hit the floor with a force equal to the weight of the leg $-\mathrm{about}$ $80 \mathrm{~N}-$ and any damage was due to a prior injury. You argue that the leg and heel dropped freely for $0.7 \mathrm{~m},$ then hit the floor and stopped in $2 \mathrm{~cm}$. What do you tell the jury about the force on the heel?

Shoukat Ali

Other Schools

Problem 92

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.20 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time.
Which of the following is greatest at the peak of the curve?
a. the ball's kinetic energy
b. the ball's speed
c. the rate at which the bat supplies energy to the ball
d. the total work the bat has done on the ball

Donald Albin

Numerade Educator

Problem 93

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.20 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time.
The ball has its maximum speed at about
a. $85 \mathrm{~ms}$.
b. $145 \mathrm{~ms}$.
c. $185 \mathrm{~ms}$.
d. whenever the force is greatest.

Donald Albin

Numerade Educator

Problem 94

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.20 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time.
As a result of being hit, the ball's kinetic energy increases by about
a. $550 \mathrm{~J}$.
b. $1.3 \mathrm{~kJ}$.
c. $7.0 \mathrm{~kJ}$.
d. You can't tell because you don't know its speed coming from the pitcher.

Donald Albin

Numerade Educator

Problem 95

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.20 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time.
The force on the ball is greatest approximately
a. at $185 \mathrm{~ms}$
b. at the peak in Fig. 6.20 .
c. before the peak in Fig. 6.20 .
d. after the peak in Fig. 6.20 but before $185 \mathrm{~ms}$.

Donald Albin

Numerade Educator