Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 7

Kinetic Energy and Work - all with Video Answers

Educators

Chapter Questions

Problem 1

A proton (mass $m=1.67 \times 10^{-27} \mathrm{~kg}$ ) is being accelerated along a straight line at $3.6 \times 10^{15} \mathrm{~m} / \mathrm{s}^{2}$ in a machine. If the proton has an initial speed of $2.4 \times 10^{7} \mathrm{~m} / \mathrm{s}$ and travels $3.5 \mathrm{~cm}$, what then is (a) its speed and (b) the increase in its kinetic energy?

Salamat Ali

Salamat Ali

Numerade Educator

Problem 2

If a Saturn $V$ rocket with an Apollo spacecraft attached had a combined mass of $2.9 \times 10^{5} \mathrm{~kg}$ and reached a speed of $11.2 \mathrm{~km} / \mathrm{s}$, how much kinetic energy would it then have?

Ze-Han Lee

Numerade Educator

Problem 3

On August 10,1972, a large meteorite skipped across the atmosphere above the western United States and western Canada, much like a stone skipped across water. The accompanying fireball was so bright that it could be seen in the daytime sky and was brighter than the usual meteorite trail. The meteorite's mass was about $4 \times 10^{6} \mathrm{~kg}$; its speed was about $15 \mathrm{~km} / \mathrm{s}$. Had it entered the atmosphere vertically, it would have hit Earth's surface with about the same speed. (a) Calculate the meteorite's loss of kinetic energy (in joules) that would have been associated with the vertical impact.
(b) Express the energy as a multiple of the explosive energy of 1 megaton of TNT, which is $4.2 \times 10^{15} \mathrm{~J}$. (c) The energy associated with the atomic bomb explosion over Hiroshima was equivalent to 13 kilotons of TNT. To how many Hiroshima bombs would the meteorite impact have been equivalent?

Prabhu Ramji

Numerade Educator

Problem 4

An explosion at ground level leaves a crater with a diameter that is proportional to the energy of the explosion raised to the $\frac{1}{3}$ power; an explosion of 1 megaton of TNT leaves a crater with a $1 \mathrm{~km}$ diameter. Below Lake Huron in Michigan there appears to be an ancient impact crater with a $50 \mathrm{~km}$ diameter. What was the kinetic energy associated with that impact, in terms of
(a) megatons of TNT (1 megaton yields $4.2 \times 10^{15} \mathrm{~J}$ ) and
(b) Hiroshima bomb equivalents (13 kilotons of TNT each)? (Ancient meteorite or comet impacts may have significantly altered the climate, killing off the dinosaurs and other life-forms.)

Prabhu Ramji

Numerade Educator

Problem 5

A father racing his son has half the kinetic energy of the son, who has half the mass of the father. The father speeds up by $1.0 \mathrm{~m} / \mathrm{s}$ and then has the same kinetic energy as the son. What are the original speeds of (a) the father and (b) the son?

Prabhu Ramji

Numerade Educator

Problem 6

A bead with mass $1.8 \times 10^{-2} \mathrm{~kg}$ is moving along a wire in the positive direction of an $x$ axis. Beginning at time $t=0$, when the bead passes through $x=0$ with speed $12 \mathrm{~m} / \mathrm{s}$, a constant force acts on the bead. Figure $7-24$ indicates the bead's position at these four times: $t_{0}=0, t_{1}=1.0 \mathrm{~s}, t_{2}=2.0 \mathrm{~s}$, and $t_{3}=3.0 \mathrm{~s}$. The bead momentarily stops at $t=3.0 \mathrm{~s}$. What is the kinetic energy of 10s?

Ze-Han Lee

Numerade Educator

Problem 7

A $3.0 \mathrm{~kg}$ body is at rest on a frictionless horizontal air track when a constant horizontal force $\vec{F}$ acting in the positive direction of an $x$ axis along the track is applied to the body. A stroboscopic graph of the position of the body as it slides to the right is shown in Fig. 7 -
25. The force $\vec{F}$ is applied to the body at $t=0$, and the graph records the position of the body at $0.50$ s intervals. How much work is done on the body by the applied force $\vec{F}$ between $t=0$ and $t=2.0 \mathrm{~s}$ ?

Prabhu Ramji

Numerade Educator

Problem 8

A ice block floating in a river is pushed through a displacement $\vec{d}=(15 \mathrm{~m}) \hat{\mathrm{i}}-(12 \mathrm{~m}) \mathrm{j}$ along a straight embankment by rushing water, which exerts a force $\vec{F}=(210 \mathrm{~N}) \mathrm{i}-(150 \mathrm{~N}) \hat{\mathrm{j}}$ on the block. How much work does the force do on the block during the displacement?

Prabhu Ramji

Numerade Educator

Problem 9

The only force acting on a $2.0 \mathrm{~kg}$ canister that is moving in an $x y$ plane has a magnitude of $5.0 \mathrm{~N}$. The canister initially has a velocity of $4.0 \mathrm{~m} / \mathrm{s}$ in the positive $x$ direction and some time later has a velocity of $6.0 \mathrm{~m} / \mathrm{s}$ in the positive $y$ direction. How much work is done on the canister by the $5.0 \mathrm{~N}$ force during this time?

Prabhu Ramji

Numerade Educator

Problem 10

A coin slides over a frictionless plane and across an $x y$ coordinate system from the origin to a point with $x y$ coordinates $(3.0 \mathrm{~m}, 4.0 \mathrm{~m})$ while a constant force acts on it. The force has magnitude $2.0 \mathrm{~N}$ and is directed at a counterclockwise angle of $100^{\circ}$ from the positive direction of the $x$ axis. How much work is done by the force on the coin during the displacement?

Ze-Han Lee

Numerade Educator

Problem 11

A $12.0 \mathrm{~N}$ force with a fixed orientation does work on a particle as the particle moves through the three-dimensional displacement $\vec{d}=(2.00 \hat{\mathrm{i}}-4.00 \hat{\mathrm{j}}+3.00 \hat{\mathrm{k}}) \mathrm{m} .$ What is the angle between the force and the displacement if the change in the particle's kinetic energy is (a) $+30.0 \mathrm{~J}$ and (b) $-30.0 \mathrm{~J}$ ?

Salamat Ali

Numerade Educator

Problem 12

A can of bolts and nuts is pushed $2.00 \mathrm{~m}$ along an $x$ axis by a broom along the greasy (frictionless) floor of a car repair shop in a version of shuffleboard. Figure $7-26$ gives the work $W$ done on the can by the constant horizontal force from the broom, versus the can's position $x .$ The scale of the figure's vertical axis is set by $W_{s}=6.0 \mathrm{~J} .(\mathrm{a})$ What is the magnitude of that force? (b) If the can had an initial kinetic energy of $3.00 \mathrm{~J}$, moving in the positive direction of the $x$ axis, what is its kinetic energy at the end of the $2.00 \mathrm{~m}$ ?

Ze-Han Lee

Numerade Educator

Problem 13

A luge and its rider, with a total mass of $85 \mathrm{~kg}$, emerge from a downhill track onto a horizontal straight track with an initial speed of $37 \mathrm{~m} / \mathrm{s}$. If a force slows them to a stop at a constant rate of $2.0$ $\mathrm{m} / \mathrm{s}^{2}$, (a) what magnitude $F$ is required for the force, (b) what distance $d$ do they travel while slowing, and (c) what work $W$ is done on them by the force? What are (d) $F,(\mathrm{e}) d$, and (f) $W$ if they, instead, slow at $4.0 \mathrm{~m} / \mathrm{s}^{2} ?$

Prabhu Ramji

Numerade Educator

Problem 14

Figure $7-27$ shows an overhead view of three horizontal forces acting on a cargo canister that was initially stationary but now moves across a frictionless floor. The force magnitudes are $F_{1}=3.00 \mathrm{~N}, F_{2}=$
$4.00 \mathrm{~N}$, and $F_{3}=10.0 \mathrm{~N}$, and the indicated angles are $\theta_{2}=50.0^{\circ}$ and $\theta_{3}=$ $35.0^{\circ}$. What is the net work done on the canister by the three forces during the first $4.00 \mathrm{~m}$ of displacement?

Ze-Han Lee

Numerade Educator

Problem 15

Figure $7-28$ shows three forces applied to a trunk that moves leftward by $3.00 \mathrm{~m}$ over a frictionless floor. The force magnitudes are $F_{1}=5.00 \mathrm{~N}, F_{2}=9.00 \mathrm{~N}$, and $F_{3}=$
$3.00 \mathrm{~N}$, and the indicated angle is $\theta=$ $60.0^{\circ}$. During the displacement,
(a) what is the net work done on the trunk by the three forces and (b) does the kinetic energy of the trunk increase or decrease?

Supratim Pal

Numerade Educator

Problem 16

An $8.0 \mathrm{~kg}$ object is moving in the positive direction of an $x$ axis. When it passes through $x=0$, a constant force directed along the axis begins to act on it. Figure $7-29$ gives its kinetic energy $K$ versus position $x$ as it moves from $x=0$ to $x=5.0 \mathrm{~m} ; K_{0}=30.0$ J. The force continues to act. What is $v$ when the object moves back through $x=-3.0 \mathrm{~m}$ ?

Nicholas Mogoi

Numerade Educator

Problem 17

A helicopter lifts a $72 \mathrm{~kg}$ astronaut $15 \mathrm{~m}$ vertically from the ocean by means of a cable. The acceleration of the astronaut is $g / 10 .$ How much work is done on the astronaut by
(a) the force from the helicopter and (b) the gravitational force on her? Just before she reaches the helicopter, what are her (c) kinetic energy and (d) speed?

James Kiss

Numerade Educator

Problem 18

(a) In 1975 the roof of Montreal's Velodrome, with a weight of $360 \mathrm{kN}$, was lifted by $10 \mathrm{~cm}$ so that it could be centered. How much work was done on the roof by the forces making the lift? (b) In 1960 a Tampa, Florida, mother reportedly raised one end of a car that had fallen onto her son when a jack failed. If her panic lift effectively raised $4000 \mathrm{~N}$ (about $\frac{1}{4}$ of the car's weight) by $5.0 \mathrm{~cm}$, how much work did her force do on the car?

Ze-Han Lee

Numerade Educator

Problem 19

In Fig. $7-30$, a block of ice slides down a frictionless ramp at angle $\theta=50^{\circ}$ while an ice worker pulls on the block (via a rope) with a force $\vec{F}_{r}$ that has a magnitude of $50 \mathrm{~N}$ and is directed up the ramp. As the block slides through distance $d=0.50 \mathrm{~m}$ along the ramp, its kinetic energy increases by 80
J. How much greater would its kinetic energy have been if the rope had not been attached to the block?

Prabhu Ramji

Numerade Educator

Problem 20

A block is sent up a frictionless ramp along which an $x$ axis extends upward. Figure $7-31$ gives the kinetic energy of the block as a function of position $x ;$ the scale of the figure's vertical axis is set by $K_{s}=40.0 \mathrm{~J}$. If the block's initial speed is $4.00 \mathrm{~m} / \mathrm{s}$, what is the normal force on the block?

Ze-Han Lee

Numerade Educator

Problem 21

A cord is used to vertically lower an initially stationary block of mass $M$ at a constant downward acceleration of $g / 4$. When the block has fallen a distance $d$, find (a) the work done by the cord's force on the block, (b) the work done by the gravitational force on the block, (c) the kinetic energy of the block, and (d) the speed of the block.

Salamat Ali

Numerade Educator

Problem 22

A cave rescue team lifts an injured spelunker directly upward and out of a sinkhole by means of a motor-driven cable. The lift is performed in three stages, each requiring a vertical distance of $10.0$ m: (a) the initially stationary spelunker is accelerated to a speed of $5.00 \mathrm{~m} / \mathrm{s} ;(\mathrm{b})$ he is then lifted at the constant speed of $5.00 \mathrm{~m} / \mathrm{s} ;$ (c) finally he is decelerated to zero speed. How much work is done on the $80.0 \mathrm{~kg}$ rescuee by the force lifting him during each stage?

Ze-Han Lee

Numerade Educator

Problem 23

In Fig. $7-32$, a constant force $\vec{F}_{a}$ of magnitude $82.0 \mathrm{~N}$ is applied to a $3.00$ $\mathrm{kg}$ shoe box at angle $\phi=53.0^{\circ}$, causing the box to move up a frictionless ramp at constant speed. How much work is done on the box by $\vec{F}_{a}$ when the box has moved through vertical distance $h=0.150 \mathrm{~m} ?$

Salamat Ali

Numerade Educator

Problem 24

In Fig. 7-33, a horizontal force $\vec{F}_{a}$ of magnitude $20.0 \mathrm{~N}$ is applied to a $3.00 \mathrm{~kg}$ psychology book as the book slides a distance $d=0.500 \mathrm{~m}$ up a frictionless ramp at angle $\theta=30.0^{\circ} .$ (a) During the displacement, what is the net work done on the book by $\vec{F}_{a}$, the gravitational force on the book, and the normal force on the book? (b) If the book has zero kinetic energy at the start of the displacement, what is its speed at the end of the displacement?

Prabhu Ramji

Numerade Educator

Problem 25

In Fig. $7-34$, a $0.250 \mathrm{~kg}$ block of cheese lies on the floor of a $900 \mathrm{~kg}$ elevator cab that is being pulled upward by a cable through distance $d_{1}=2.40 \mathrm{~m}$ and then through distance $d_{2}=10.5 \mathrm{~m} .$ (a) Through $d_{1}$, if the normal force on the block from the floor has constant magnitude $F_{N}=3.00 \mathrm{~N}$, how much work is done on the cab by the force from the cable? (b) Through $d_{2}$, if the work done on the cab by the (constant) force from the cable is $92.61 \mathrm{~kJ}$, what is the magnitude of $F_{N} ?$

Salamat Ali

Numerade Educator

Problem 26

In Fig. 7-10, we must apply a force of magnitude $80 \mathrm{~N}$ to hold the block stationary at $x=-2.0 \mathrm{~cm}$. From that position, we then slowly move the block so that our force does $+4.0 \mathrm{~J}$ of work on the spring-block system; the block is then again stationary. What is the block's position? (Hint: There are two answers)

Ze-Han Lee

Numerade Educator

Problem 27

A spring and block are in the arrangement of Fig. $7-10$. When the block is pulled out to $x=+4.0 \mathrm{~cm}$, we must apply a force of magnitude $360 \mathrm{~N}$ to hold it there. We pull the block to $x=11 \mathrm{~cm}$ and then release it. How much work does the spring do on the block as the block moves from $x_{i}=+5.0 \mathrm{~cm}$ to (a) $x=+3.0 \mathrm{~cm}$, (b) $x=-3.0 \mathrm{~cm}$,
(c) $x=-5.0 \mathrm{~cm}$, and $($ d $) x=-9.0 \mathrm{~cm}$ ?

Salamat Ali

Numerade Educator

Problem 28

During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one another with large catapults that are made with surgical hose mounted on a window frame. A balloon filled with dyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room. Assume that the stretching of the hose obeys Hooke's law with a spring constant of $100 \mathrm{~N} / \mathrm{m}$. If the hose is stretched by $5.00 \mathrm{~m}$ and then released, how much work does the force from the hose do on the balloon in the pouch by the time the hose reaches its relaxed length?

Ze-Han Lee

Numerade Educator

Problem 29

In the arrangement of Fig. 7-10, we gradually pull the block from $x=0$ to $x=+3.0 \mathrm{~cm}$, where it is stationary. Figure $7-35$ gives the work that our force does on the block. The scale of the figure's vertical axis is set by $W_{s}=1.0 \mathrm{~J}$. We then pull the block out to $x=$ $+5.0 \mathrm{~cm}$ and release it from rest. How much work does the spring do on the block when the block moves from $x_{i}=+5.0 \mathrm{~cm}$ to
(a) $x=+4.0 \mathrm{~cm}$,
(b) $x=-2.0 \mathrm{~cm}$, and
(c) $x=-5.0 \mathrm{~cm}$ ?

Salamat Ali

Numerade Educator

Problem 30

In Fig. $7-10 a$, a block of mass $m$ lies on a horizontal frictionless surface and is attached to one end of a horizontal spring (spring constant $k$ ) whose other end is fixed. The block is initially at rest at the position where the spring is unstretched $(x=0)$ when a constant horizontal force $\vec{F}$ in the positive direction of the $x$ axis is applied to it. A plot of the resulting kinetic energy of the block versus its position $x$ is shown in Fig. $7-36 .$ The scale of the figure's vertical axis is set by $K_{s}=4.0 \mathrm{~J} .(\mathrm{a})$ What is the magnitude of $\vec{F} ?(\mathrm{~b})$ What is the value of $k ?$

Ze-Han Lee

Numerade Educator

Problem 31

The only force acting on a $2.0 \mathrm{~kg}$ body as it moves along a positive $x$ axis has an $x$ component $F_{x}=-6 x \mathrm{~N}$, with $x$ in meters. The velocity at $x=3.0 \mathrm{~m}$ is $8.0 \mathrm{~m} / \mathrm{s}$. (a) What is the velocity of the body at $x=4.0 \mathrm{~m} ?$ (b) At what positive value of $x$ will the body have a velocity of $5.0 \mathrm{~m} / \mathrm{s}$ ?

Salamat Ali

Numerade Educator

Problem 32

Figure 7-37 gives spring force $F_{x}$ versus position $x$ for the spring-block arrangement of Fig. 7- 10. The scale is set by $F_{s}=160.0 \mathrm{~N}$. We release the block at $x=12 \mathrm{~cm}$. How much work does the spring do on the block when the block moves from $x_{i}=+8.0 \mathrm{~cm}$ to (a) $x=+5.0$ $\mathrm{cm},($ b) $x=-5.0 \mathrm{~cm},(\mathrm{c}) x=-8.0$ $\mathrm{cm}$, and $(\mathrm{d}) x=-10.0 \mathrm{~cm} ?$

Ze-Han Lee

Numerade Educator

Problem 33

The block in Fig. $7-10 a$ lies on a horizontal frictionless surface, and the spring constant is $50 \mathrm{~N} / \mathrm{m}$. Initially, the spring is at its relaxed length and the block is stationary at position $x=0 .$ Then an applied force with a constant magnitude of $3.0 \mathrm{~N}$ pulls the block in the positive direction of the $x$ axis, stretching the spring until the block stops. When that stopping point is reached, what are
(a) the position of the block, (b) the work that has been done on the block by the applied force, and (c) the work that has been done on the block by the spring force? During the block's displacement, what are (d) the block's position when its kinetic energy is maximum and (e) the value of that maximum kinetic energy?

Salamat Ali

Numerade Educator

Problem 34

A $10 \mathrm{~kg}$ brick moves along an $x$ axis. Its acceleration as a function of its position is shown in Fig. $7-38$. The scale of the figure's vertical axis is set by $a_{s}=20.0 \mathrm{~m} / \mathrm{s}^{2}$. What is the net work performed on the brick by the force causing the acceleration as the brick moves from $x=0$ to $x=8.0 \mathrm{~m} ?$

Prabhu Ramji

Numerade Educator

Problem 35

The force on a particle is directed along an $x$ axis and given by $F=F_{0}\left(x / x_{0}-1\right) .$ Find the work done by the force in moving the particle from $x=0$ to $x=2 x_{0}$ by (a) plotting $F(x)$ and measuring the work from the graph and (b) integrating $F(x)$.

Salamat Ali

Numerade Educator

Problem 36

A $5.0 \mathrm{~kg}$ block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as shown in Fig. $7-39 .$ The scale of the figure's vertical axis is set by $F_{s}=10.0 \mathrm{~N}$. How much work is done by the force as the block moves from the origin to $x=8.0 \mathrm{~m}$ ?

Ze-Han Lee

Numerade Educator

Problem 37

Figure $7-40$ gives the acceleration of a $2.00 \mathrm{~kg}$ particle as an applied force $\vec{F}_{a}$ moves it from rest along an $x$ axis from $x=0$ to $x=9.0 \mathrm{~m}$. The scale of the figure's vertical axis is set by $a_{s}=6.0 \mathrm{~m} / \mathrm{s}^{2}$. How much work has the force done on the particle when the particle reaches (a) $x=4.0 \mathrm{~m}$, (h) $x=7.0 \mathrm{~m}$, and (c) $x=9.0 \mathrm{~m}$ ? What is the particle's speed and direction of travel when it reaches (d) $x=4.0 \mathrm{~m},(\mathrm{e}) x=7.0 \mathrm{~m}$, and
(f) $x=9.0 \mathrm{~m}$ ?

Donald Albin

Numerade Educator

Problem 38

A $1.5 \mathrm{~kg}$ block is initially at rest on a horizontal frictionless surface when a horizontal force along an $x$ axis is applied to the block. The force is given by $\vec{F}(x)=\left(2.5-x^{2}\right) \hat{\mathrm{i}} \mathrm{N}$, where $x$ is in meters and the initial position of the block is $x=0 .$ (a) What is the kinetic energy of the block as it passes through $x=2.0 \mathrm{~m} ?$ (b) What is the maximum kinetic energy of the block between $x=0$ and $x=2.0 \mathrm{~m}$ ?

Prabhu Ramji

Numerade Educator

Problem 39

A force $\vec{F}=\left(c x-3.00 x^{2}\right) \hat{\mathrm{i}}$ acts on a particle as the particle moves along an $x$ axis, with $\vec{F}$ in newtons, $x$ in meters, and $c$ a constant. At $x=0$, the particle's kinetic energy is $20.0 \mathrm{~J} ;$ at $x=3.00 \mathrm{~m}$, it is $11.0 \mathrm{~J}$. Find $c$.

Prabhu Ramji

Numerade Educator

Problem 40

A can of sardines is made to move along an $x$ axis from $x=0.25 \mathrm{~m}$ to $x=1.25 \mathrm{~m}$ by a force with a magnitude given by $F=\exp \left(-4 x^{2}\right)$, with $x$ in meters and $F$ in newtons (Here exp is the exponential function.) How much work is done on the can by the force?

Donald Albin

Numerade Educator

Problem 41

A single force acts on a $3.0 \mathrm{~kg}$ particle-like object whose position is given by $x=3.0 t-4.0 t^{2}+1.0 t^{3}$, with $x$ in meters and $t$ in seconds. Find the work done by the force from $t=0$ to $t=4.0 \mathrm{~s}$.

Salamat Ali

Numerade Educator

Problem 42

Figure $7-41$ shows a cord attached to a cart that can slide along a frictionless horizontal rail aligned along an $x$ axis. The left end of the cord is pulled over a pulley, of negligible mass and friction and at cord height $h=1.20 \mathrm{~m}$, so the cart slides from $x_{1}=3.00 \mathrm{~m}$ to $x_{2}=1.00 \mathrm{~m}$. During the move, the tension in the cord is a constant $25.0 \mathrm{~N}$. What is the change in the kinetic energy of the cart during the move?

Ze-Han Lee

Numerade Educator

Problem 43

A force of $5.0 \mathrm{~N}$ acts on a $15 \mathrm{~kg}$ body initially at rest. Compute the work done by the force in (a) the first, (b) the second, and (c) the third seconds and (d) the instantaneous power due to the force at the end of the third second.

Salamat Ali

Numerade Educator

Problem 44

A skier is pulled by a towrope up a frictionless ski slope that makes an angle of $12^{\circ}$ with the horizontal. The rope moves parallel to the slope with a constant speed of $1.0 \mathrm{~m} / \mathrm{s}$. The force of the rope does $900 \mathrm{~J}$ of work on the skier as the skier moves a distance of $8.0 \mathrm{~m}$ up the incline. (a) If the rope moved with a constant speed of $2.0 \mathrm{~m} / \mathrm{s}$, how much work would the force of the rope do on the skier as the skier moved a distance of $8.0 \mathrm{~m}$ up the incline? At what rate is the force of the rope doing work on the skier when the rope moves with a speed of (b) $1.0 \mathrm{~m} / \mathrm{s}$ and
(c) $2.0 \mathrm{~m} / \mathrm{s}$ ?

Ze-Han Lee

Numerade Educator

Problem 45

A $100 \mathrm{~kg}$ block is pulled at a constant speed of $5.0 \mathrm{~m} / \mathrm{s}$ across a horizontal floor by an applied force of $122 \mathrm{~N}$ directed $37^{\circ}$ above the horizontal. What is the rate at which the force does work on the block?

Salamat Ali

Numerade Educator

Problem 46

The loaded cab of an elevator has a mass of $3.0 \times 10^{3} \mathrm{~kg}$ and moves $210 \mathrm{~m}$ up the shaft in $23 \mathrm{~s}$ at constant speed. At what average rate does the force from the cable do work on the cab?

Vishal Gupta

Numerade Educator

Problem 47

A machine carries a $4.0 \mathrm{~kg}$ package from an initial position of $\vec{d}_{i}=(0.50 \mathrm{~m}) \hat{\mathrm{i}}+(0.75 \mathrm{~m}) \hat{\mathrm{j}}+(0.20 \mathrm{~m}) \hat{\mathrm{k}}$ at $t=0$ to a final posi-
tion of $\vec{d}_{f}=(7.50 \mathrm{~m}) \hat{\mathrm{i}}+(12.0 \mathrm{~m}) \hat{\mathrm{j}}+(7.20 \mathrm{~m}) \hat{\mathrm{k}}$ at $t=12 \mathrm{~s}$. The constant force applied by the machine on the package is $\vec{F}=(2.00 \mathrm{~N}) \hat{\mathrm{i}}+(4.00 \mathrm{~N}) \hat{\mathrm{j}}+(6.00 \mathrm{~N}) \hat{\mathrm{k}}$. For that displacement, find (a) the work done on the package by the machine's force and (b) the average power of the machine's force on the package.

Prabhu Ramji

Numerade Educator

Problem 48

A $0.30 \mathrm{~kg}$ ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring $(k=500 \mathrm{~N} / \mathrm{m})$ whose other end is fixed. The ladle has a kinetic energy of $10 \mathrm{~J}$ as it passes through its equilibrium position (the point at which the spring force is zero). (a) At what rate is the spring doing work on the ladle as the ladle passes through its equilibrium position?
(b) At what rate is the spring doing work on the ladle when the spring is compressed $0.10 \mathrm{~m}$ and the ladle is moving away from the equilibrium position?

Ze-Han Lee

Numerade Educator

Problem 49

A fully loaded, slow-moving freight elevator has a cab with a total mass of $1200 \mathrm{~kg}$, which is required to travel upward $54 \mathrm{~m}$ in $3.0$ min, starting and ending at rest. The elevator's counterweight has a mass of only $950 \mathrm{~kg}$, and so the elevator motor must help. What average power is required of the force the motor exerts on the cab via the cable?

Prabhu Ramji

Numerade Educator

Problem 50

(a) At a certain instant, a particle-like object is acted on by a force $\vec{F}=(4.0 \mathrm{~N}) \hat{\mathrm{i}}-(2.0 \mathrm{~N}) \hat{\mathrm{j}}+(9.0 \mathrm{~N}) \hat{\mathrm{k}}$ while the object's veloc-
ity is $\vec{v}=-(2.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(4.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{k}}$. What is the instantaneous rate at which the force does work on the object? (b) At some other time, the velocity consists of only a $y$ component. If the force is unchanged and the instantaneous power is $-12 \mathrm{~W}$, what is the velocity of the object?

Prabhu Ramji

Numerade Educator

Problem 51

A force $\vec{F}=(3.00 \mathrm{~N}) \hat{\mathrm{i}}+(7.00 \mathrm{~N}) \hat{\mathrm{j}}+(7.00 \mathrm{~N}) \hat{\mathrm{k}}$ acts on a $2.00 \mathrm{~kg}$ mobile object that moves from an initial position of $\vec{d}_{i}=(3.00 \mathrm{~m}) \hat{\mathrm{i}}-(2.00 \mathrm{~m}) \hat{\mathrm{j}}+(5.00 \mathrm{~m}) \hat{\mathrm{k}}$ to a final position of $\vec{d}_{f}=-(5.00 \mathrm{~m}) \hat{\mathrm{i}}+(4.00 \mathrm{~m}) \hat{\mathrm{j}}+(7.00 \mathrm{~m}) \hat{\mathrm{k}}$ in $4.00 \mathrm{~s}$. Find (a) the work done on the object by the force in the $4.00 \mathrm{~s}$ interval, (b) the average power due to the force during that interval, and (c) the angle between vectors $\vec{d}_{i}$ and $\vec{d}_{f}$.

Sheh Lit Chang

University of Washington

Problem 52

A funny car accelerates from rest through a measured track distance in time $T$ with the engine operating at a constant power $P$. If the track crew can increase the engine power by a differential amount $d P$, what is the change in the time required for the run?

Ze-Han Lee

Numerade Educator

Problem 53

Figure $7-42$ shows a cold package of hot dogs sliding rightward across a frictionless floor through a distance $d=20.0 \mathrm{~cm}$ while three forces act on the package. Two of them are horizontal and have the magnitudes $F_{1}=5.00 \mathrm{~N}$ and $F_{2}=1.00 \mathrm{~N} ;$ the third is angled down by $\theta=60.0^{\circ}$ and has the magnitude $F_{3}=4.00 \mathrm{~N}$.
(a) For the $20.0 \mathrm{~cm}$ displacement, what is the net work done on the package by the three applied forces, the gravitational force on the package, and the normal force on the package? (b) If the package has a mass of $2.0 \mathrm{~kg}$ and an initial kinetic energy of 0, what is its speed at the end of the displacement?

Prabhu Ramji

Numerade Educator

Problem 54

The only force acting on a $2.0 \mathrm{~kg}$ body as the body moves along an $x$ axis varies as shown in Fig. $7-43$. The scale of the figure's vertical axis is set by $F_{s}=4.0 \mathrm{~N}$. The velocity of the body at $x=0$ is $4.0 \mathrm{~m} / \mathrm{s}$. (a) What is the kinetic energy of the body at $x=3.0 \mathrm{~m} ?$ (b) At what value of $x$ will the body have a kinetic energy of $8.0 \mathrm{~J} ?$ (c) What is the maximum kinetic energy of the body between $x=0$ and $x=5.0 \mathrm{~m}$ ?

Narayan Hari

Numerade Educator

Problem 55

A horse pulls a cart with a force of $40 \mathrm{lb}$ at an angle of $30^{\circ}$ above the horizontal and moves along at a speed of $6.0 \mathrm{mi} / \mathrm{h} .$ (a) How much work does the force do in $10 \mathrm{~min} ?$ (b) What is the average power (in horsepower) of the force?

Prabhu Ramji

Numerade Educator

Problem 56

An initially stationary $2.0 \mathrm{~kg}$ object accelerates horizontally and uniformly to a speed of $10 \mathrm{~m} / \mathrm{s}$ in $3.0 \mathrm{~s}$. (a) In that $3.0 \mathrm{~s}$ interval, how much work is done on the object by the force accelerating it? What is the instantaneous power due to that force (b) at the end of the interval and (c) at the end of the first half of the interval?

Ze-Han Lee

Numerade Educator

Problem 57

A $230 \mathrm{~kg}$ crate hangs from the end of a rope of length $L=12.0 \mathrm{~m}$. You push horizontally on the crate with a varying force $\vec{F}$ to move it distance $d=$ $4.00 \mathrm{~m}$ to the side (Fig. 7-44). (a) What is the magnitude of $\vec{F}$ when the crate is in this final position? During the crate's displacement, what are (b) the total work done on it, (c) the work done by the gravitational force on the crate, and (d) the work done by the pull on the crate from the rope?
(e) Knowing that the crate is motionless before and after its displacement, use the answers to (b), (c), and (d) to find the work your force $\vec{F}$ does on the crate. (f) Why is the work of your force not equal to the product of the horizontal displacement and the answer to (a)?

Salamat Ali

Numerade Educator

Problem 58

To pull a $50 \mathrm{~kg}$ crate across a horizontal frictionless floor, a worker applies a force of $210 \mathrm{~N}$, directed $20^{\circ}$ above the horizontal. As the crate moves $3.0 \mathrm{~m}$, what work is done on the crate by (a) the worker's force, (b) the gravitational force, and (c) the normal force? (d) What is the total work?

Ze-Han Lee

Numerade Educator

Problem 59

A force $\vec{F}_{a}$ is applied to a bead as the bead is moved along a straight wire through displacement $+5.0 \mathrm{~cm}$. The magnitude of $\vec{F}_{a}$ is set at a certain value, but the angle $\phi$ between $\vec{F}_{a}$ and the bead's displacement can be chosen. Figure $7-45$ gives the work $W$ done by $\vec{F}_{a}$ on the bead for a range of $\phi$ values; $W_{0}=25 \mathrm{~J}$. How much work is done by $\vec{F}_{a}$ if $\phi$ is (a) $64^{\circ}$ and (b) $147^{\circ}$ ?

Salamat Ali

Numerade Educator

Problem 60

A frightened child is restrained by her mother as the child slides down a frictionless playground slide. If the force on the child from the mother is $100 \mathrm{~N}$ up the slide, the child's kinetic energy increases by $30 \mathrm{~J}$ as she moves down the slide a distance of $1.8 \mathrm{~m}$. (a) How much work is done on the child by the gravitational force during the $1.8 \mathrm{~m}$ descent?
(b) If the child is not restrained by her mother, how much will the child's kinetic energy increase as she comes down the slide that same distance of $1.8 \mathrm{~m} ?$

Ze-Han Lee

Numerade Educator

Problem 61

How much work is done by a force $\vec{F}=(2 x \mathrm{~N}) \hat{\mathrm{i}}+(3 \mathrm{~N}) \hat{\mathrm{j}}$,
with $x$ in meters, that moves a particle from a position $\vec{r}_{i}=$ $(2 \mathrm{~m}) \hat{\mathrm{i}}+(3 \mathrm{~m}) \hat{\mathrm{j}}$ to a position $\vec{r}_{f}=-(4 \mathrm{~m}) \hat{\mathrm{i}}-(3 \mathrm{~m}) \hat{\mathrm{j}}$ ?

Salamat Ali

Numerade Educator

Problem 62

A $250 \mathrm{~g}$ block is dropped onto a relaxed vertical spring that has a spring constant of $k=$ $2.5 \mathrm{~N} / \mathrm{cm}$ (Fig. $7-46)$. The block becomes attached to the spring and compresses the spring $12 \mathrm{~cm}$ before momentarily stopping. While the spring is being compressed, what work is done on the block by (a) the gravitational force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring? (Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum compression of the spring?

Ze-Han Lee

Numerade Educator

Problem 63

To push a $25.0 \mathrm{~kg}$ crate up a frictionless incline, angled at $25.0^{\circ}$ to the horizontal, a worker exerts a force of $209 \mathrm{~N}$ parallel to the incline. As the crate slides $1.50 \mathrm{~m}$, how much work is done on the crate by (a) the worker's applied force, (b) the gravitational force on the crate, and (c) the normal force exerted by the incline on the crate? (d) What is the total work done on the crate?

Salamat Ali

Numerade Educator

Problem 64

Boxes are transported from one location to another in a warehouse by means of a conveyor belt that moves with a constant speed of $0.50 \mathrm{~m} / \mathrm{s}$. At a certain location the conveyor belt moves for $2.0 \mathrm{~m}$ up an incline that makes an angle of $10^{\circ}$ with the horizontal, then for $2.0 \mathrm{~m}$ horizontally, and finally for $2.0 \mathrm{~m}$ down an incline that makes an angle of $10^{\circ}$ with the horizontal. Assume that a $2.0 \mathrm{~kg}$ box rides on the belt without slipping. At what rate is the force of the conveyor belt doing work on the box as the box moves (a) up the $10^{\circ}$ incline, (b) horizontally, and (c) down the $10^{\circ}$ incline?

Ze-Han Lee

Numerade Educator

Problem 65

In Fig. $7-47$, a cord runs around two massless, frictionless pulleys. A canister with mass $m=20 \mathrm{~kg}$ hangs from one pulley, and you exert a force $\vec{F}$ on the free end of the cord.
(a) What must be the magnitude of $\vec{F}$ if you are to lift the canister at a constant speed? (b) To lift the canister by $2.0 \mathrm{~cm}$, how far must you pull the free end of the cord? During that lift, what is the work done on the canister by (c) your force (via the cord) and
(d) the gravitational force? (Hint:
When a cord loops around a pulley as shown. it pulls on the pullev with a net force that is twice the tension in the cord.)

Prabhu Ramji

Numerade Educator

Problem 66

If a car of mass $1200 \mathrm{~kg}$ is moving along a highway at $120 \mathrm{~km} / \mathrm{h}$, what is the car's kinetic energy as determined by someone standing alongside the highway?

Ze-Han Lee

Numerade Educator

Problem 67

A spring with a pointer attached is hanging next to a scale marked in millimeters. Three different packages are hung from the spring, in turn, as shown in Fig. $7-48$. (a) Which mark on the scale will the pointer indicate when no package is hung from the spring? (b) What is the weight $W$ of the third package?

Salamat Ali

Numerade Educator

Problem 68

An iceboat is at rest on a frictionless frozen lake when a sudden wind exerts a constant force of $200 \mathrm{~N}$, toward the east, on the boat. Due to the angle of the sail, the wind causes the boat to slide in a straight line for a distance of $8.0 \mathrm{~m}$ in a direction $20^{\circ}$ north of east. What is the kinetic energy of the iceboat at the end of that $8.0 \mathrm{~m} ?$

Prabhu Ramji

Numerade Educator

Problem 69

If a ski lift raises 100 passengers averaging $660 \mathrm{~N}$ in weight to a height of $150 \mathrm{~m}$ in $60.0 \mathrm{~s}$, at constant speed, what average power is required of the force making the lift?

Salamat Ali

Numerade Educator

Problem 70

A force $\vec{F}=(4.0 \mathrm{~N}) \hat{\mathrm{i}}+c \hat{\mathrm{j}}$ acts on a particle as the particle goes through displacement $\vec{d}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}-(2.0 \mathrm{~m}) \hat{\mathrm{j}} .$ (Other forces also act on the particle.) What is $c$ if the work done on the particle by force $\vec{F}$ is (a) 0, (b) $17 \mathrm{~J}$, and (c) $-18 \mathrm{~J}$ ?

Ze-Han Lee

Numerade Educator

Problem 71

A constant force of magnitude $10 \mathrm{~N}$ makes an angle of $150^{\circ}$ (measured counterclockwise) with the positive $x$ direction as it acts on a $2.0 \mathrm{~kg}$ object moving in an $x y$ plane. How much work is done on the object by the force as the object moves from the origin to the point having position vector $(2.0 \mathrm{~m}) \hat{\mathrm{i}}-(4.0 \mathrm{~m}) \hat{\mathrm{j}}$ ?

Salamat Ali

Numerade Educator

Problem 72

In Fig. $7-49 a$, a $2.0 \mathrm{~N}$ force is applied to a $4.0 \mathrm{~kg}$ block at a downward angle $\theta$ as the block moves rightward through $1.0 \mathrm{~m}$ across a frictionless floor. Find an expression for the speed $v_{f}$ of the block at the end of that distance if the block's initial velocity is
(a) 0 and (b) $1.0 \mathrm{~m} / \mathrm{s}$ to the right. (c) The situation in Fig. $7-49 b$ is similar in that the block is initially moving at $1.0 \mathrm{~m} / \mathrm{s}$ to the right, but now the $2.0 \mathrm{~N}$ force is directed downward to the left. Find an expression for the speed $v_{f}$ of the block at the end of the $1.0 \mathrm{~m}$ distance. (d) Graph all three expressions for $v_{f}$ versus downward angle $\theta$ for $\theta=0^{\circ}$ to $\theta=90^{\circ}$. Interpret the graphs.

Donald Albin

Numerade Educator

Problem 73

A force $\vec{F}$ in the positive direction of an $x$ axis acts on an object moving along the axis. If the magnitude of the force is $F=10 e^{-\mathrm{v} 20}$ $\mathrm{N}$, with $x$ in meters, find the work done by $\vec{F}$ as the object moves from $x=0$ to $x=2.0 \mathrm{~m}$ by (a) plotting $F(x)$ and estimating the area under the curve and (b) integrating to find the work analytically.

Donald Albin

Numerade Educator

Problem 74

A particle moves along a straight path through displacement $\vec{d}=(8 \mathrm{~m}) \hat{\mathrm{i}}+c \hat{\mathrm{j}}$ while force $\vec{F}=(2 \mathrm{~N}) \hat{\mathrm{i}}-(4 \mathrm{~N}) \hat{\mathrm{j}}$ acts on it. (Other
forces also act on the particle.) What is the value of $c$ if the work done by $\vec{F}$ on the particle is (a) zero, (b) positive, and (c) negative?

Ze-Han Lee

Numerade Educator

Problem 75

What is the power of the force required to move a 4500 $\mathrm{kg}$ elevator cab with a load of $1800 \mathrm{~kg}$ upward at constant speed $3.80 \mathrm{~m} / \mathrm{s} ?$

Salamat Ali

Numerade Educator

Problem 76

A $45 \mathrm{~kg}$ block of ice slides down a frictionless incline $1.5 \mathrm{~m}$ long and $0.91 \mathrm{~m}$ high. A worker pushes up against the ice, parallel to the incline, so that the block slides down at constant speed.
(a) Find the magnitude of the worker's force. How much work is done on the block by (b) the worker's force, (c) the gravitational force on the block, (d) the normal force on the block from the surface of the incline, and (e) the net force on the block?

Ze-Han Lee

Numerade Educator

Problem 77

As a particle moves along an $x$ axis, a force in the positive direction of the axis acts on it. Figure $7-50$ shows the magnitude $F$ of the force versus position $x$ of the particle. The curve is given by $F=a / x^{2}$, with $a=9.0 \mathrm{~N} \cdot \mathrm{m}^{2}$. Find the work done on the particle by the force as the particle moves from $x=1.0 \mathrm{~m}$ to $x=3.0 \mathrm{~m}$ by (a) estimating the work from the graph and (b) integrating the force function.

Salamat Ali

Numerade Educator

Problem 78

A CD case slides along a floor in the positive direction of an $x$ axis while an applied force $\vec{F}_{a}$ acts on the case. The force is directed along the $x$ axis and has the $x$ component $F_{a x}=9 x-3 x^{2}$, with $x$ in meters and $F_{a x}$ in newtons. The case starts at rest at the position $x=0$, and it moves until it is again at rest. (a) Plot the work $\vec{F}_{d}$ does on the case as a function of $x$. (b) At what position is the work maximum, and (c) what is that maximum value? (d) At what position has the work decreased to zero? (e) At what position is the case again at rest?

Ze-Han Lee

Numerade Educator

Problem 79

A $2.0 \mathrm{~kg}$ lunchbox is sent sliding over a frictionless surface, in the positive direction of an $x$ axis along the surface. Beginning at time $t=0$, a steady wind pushes on the lunchbox in the negative direction of the $x$ axis. Figure $7-51$ shows the position $x$ of the lunchbox as a function of time $t$ as the wind pushes on the lunchbox. From the graph, estimate the kinetic energy of the lunchbox at
(a) $t=1.0 \mathrm{~s}$ and
(b) $t=5.0 \mathrm{~s}$.
(c) How much work does the force from the wind do on the lunchbox from $t=1.0 \mathrm{~s}$ to $t=5.0 \mathrm{~s}$ ?

Donald Albin

Numerade Educator

Problem 80

A breadbox is made to move along an $x$ axis from $x=0.15 \mathrm{~m}$ to $x=1.20 \mathrm{~m}$ by a force with a magnitude given by $F=\exp \left(-2 x^{2}\right)$, with $x$ in meters and $F$ in newtons. (Here exp is the exponential function.) How much work is done on the breadbox by the force?

Donald Albin

Numerade Educator

Problem 81

In the block-spring arrangement of Fig. $7-10$, the block's mass is $4.00 \mathrm{~kg}$ and the spring constant is $500 \mathrm{~N} / \mathrm{m}$. The block is released from position $x_{i}=0.300 \mathrm{~m}$. What are (a) the block's speed at $x=0$,
(b) the work done by the spring when the block reaches $x=0,(\mathrm{c})$ the instantaneous power due to the spring at the release point $x_{i}$,
(d) the instantaneous power at $x=0$, and (e) the block's position when the power is maximum?

Salamat Ali

Numerade Educator

Problem 82

A $4.00 \mathrm{~kg}$ block is pulled up a frictionless inclined plane by a $50.0 \mathrm{~N}$ force that is parallel to the plane, starting from rest. The normal force on the block from the plane has magnitude $13.41 \mathrm{~N}$. What is the block's speed when its displacement up the ramp is $3.00 \mathrm{~m}$ ?

Ze-Han Lee

Numerade Educator

Problem 83

A spring with a spring constant of $18.0 \mathrm{~N} / \mathrm{cm}$ has a cage attached to its free end. (a) How much work does the spring force do on the cage when the spring is stretched from its relaxed length by $7.60 \mathrm{~mm} ?$ (b) How much additional work is done by the spring force when the spring is stretched by an additional $7.60 \mathrm{~mm} ?$

Salamat Ali

Numerade Educator

Problem 84

A force $\vec{F}=(2.00 \hat{1}+9.00 \hat{\mathrm{j}}+5.30 \hat{\mathrm{k}}) \mathrm{N}$ acts on a $2.90 \mathrm{~kg}$
object that moves in time interval $2.10 \mathrm{~s}$ from an initial position $\vec{r}_{1}=(2.70 \hat{i}-2.90 \hat{j}+5.50 \hat{k}) \mathrm{m}$ to a final position $\overrightarrow{\vec{r}}_{2}=$
$(-4.10 \hat{\mathrm{i}}+3.30 \hat{\mathrm{j}}+5.40 \mathrm{k}) \mathrm{m}$. Find (a) the work done on the object
by the force in that time interval, (b) the average power due to the force during that time interval, and (c) the angle between vectors $\vec{r}_{1}$ and $\vec{r}_{2}$.

Prabhu Ramji

Numerade Educator

Problem 85

At $t=0$, force $\vec{F}=(-5.00 \hat{\mathrm{i}}+5.00 \hat{\mathrm{j}}+4.00 \mathrm{k}) \mathrm{N}$ begins to act
on a $2.00 \mathrm{~kg}$ particle with an initial speed of $4.00 \mathrm{~m} / \mathrm{s}$. What is the particle's speed when its displacement from the initial point is $\vec{d}=(2.00 \hat{1}+2.00 \hat{j}+7.00 \hat{k}) \mathrm{m} ?$

Prabhu Ramji

Numerade Educator