Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 7

Kinetic Energy and Work - all with Video Answers

Educators

Chapter Questions

Problem 1

When accelerated along a straight line at $2.8 \times 10^{15} \mathrm{~m} / \mathrm{s}^{2}$ in a machine, an electron (mass $m=9.1 \times 10^{-31} \mathrm{~kg}$ ) has an initial speed of $1.4 \times 10^{7} \mathrm{~m} / \mathrm{s}$ and travels $5.8 \mathrm{~cm}$. Find (a) the final speed of the electron and (b) the increase in its kinetic energy.

Sunita Kumari

Sunita Kumari

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

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 7-16 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 $\phi$ \text { is (a) } 64^{\circ} \text { and (b) } 147^{\circ} \text { ? }

Salamat Ali

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 7-17 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 the bead at $t=10 \mathrm{~s}$ ?
Figure 7-17 Problem $6 .$

Sunita Kumari

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. 718. The force $\vec{F}$ is applied to the body at $t=0$, and the graph records the position of the body at $0.50 \mathrm{~s}$ intervals. How much work is done on the body by the applied force $\vec{F}$ between $t=0$ and $t=2.0 \mathrm{~s}$ ?
Figure 7-18 Problem 7.

Sunita Kumari

Numerade Educator

Problem 8

A ice block floating in a river is pushed through a displacement $\vec{d}=(20 \mathrm{~m}) \hat{\mathrm{i}}-(16 \mathrm{~m}) \hat{\mathrm{j}}$ along a straight embankment by rushing water, which exerts a force $\vec{F}=(210 \mathrm{~N}) \hat{\mathrm{i}}-(150 \mathrm{~N}) \hat{\mathrm{j}}$ on the block. How much work does the force do on the block during the displacement?

Sunita Kumari

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 $2.5 \mathrm{~N}$ 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?

Sunita Kumari

Numerade Educator

Problem 11

A particle travels through a three-dimensional displacement given by $\vec{d}=(5.00 \hat{\mathrm{i}}-3.00 \hat{\mathrm{j}}+4.00 \hat{\mathrm{k}}) \mathrm{m}$. If a force of magnitude $22.0 \mathrm{~N}$ and with fixed orientation does work on the particle, find the angle between the force and the displacement if the change in the particle's kinetic energy is (a) $45.0 \mathrm{~J}$ and (b) $-45.0 \mathrm{~J}$.

Sunita Kumari

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 $7-19$ 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 $W_{s}=6.0 \mathrm{~J}$. (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) $F$, (e) $d$, and (f) $W$ if they, instead, slow at $4.0 \mathrm{~m} / \mathrm{s}^{2}$ ?

Prabhu Ramji

Numerade Educator

Problem 14

Figure 7-20 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 $F_{3}=9.00 \mathrm{~N}$, 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?

Sunita Kumari

Numerade Educator

Problem 15

Figure 7-21 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?

Sunita Kumari

Numerade Educator

Problem 16

A $7.0 \mathrm{~kg}$ object is moving in $K(\mathrm{~J})$
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 $7-22$
gives its kinetic energy $K$ versus
position $x$ as it moves from $x=0$ to
$x=5.0 \mathrm{~m} ; K_{0}=30.0 \mathrm{~J}$. The force

Sunita Kumari

Numerade Educator

Problem 17

A military helicopter lifts a $75 \mathrm{~kg}$ flood survivor $16 \mathrm{~m}$ vertically from the river by a rope. If the acceleration of the survivor is $\mathrm{g} / 10$, how much work is done on the survivor 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?

Sunita Kumari

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. 7-23, 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 7-24 gives the kinetic energy of the block as a function of position $x$; the scale of the figure's vertical axis is set by $K_{s}=50.0 \mathrm{~J}$. If the block's initial speed is $5.00 \mathrm{~m} / \mathrm{s}$, what is the normal force on the block?

Narayan Hari

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 $12.0$ $\mathrm{m}$ : (a) the initially stationary spelunker is accelerated to a speed of $5.00 \mathrm{~m} / \mathrm{s} ;$ (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 $85.0 \mathrm{~kg}$ rescuee by the force lifting him during each stage?

Neelesh Sharma

Numerade Educator

Problem 23

In Fig. 7-25, 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. 7-26, a horizontal force $\vec{F}_{a}$ of magnitude $23.0 \mathrm{~N}$ is applied to a $3.00 \mathrm{~kg}$ psychology book as the book slides a distance $d=0.580 \mathrm{~m}$ 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?

Sunita Kumari

Numerade Educator

Problem 25

In Fig. 7-27, 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

A spring of spring constant $5.0 \times 10^{3} \mathrm{~N} / \mathrm{m}$ is stretched initially by $5.0 \mathrm{~cm}$ from the unstretched position. What is the work required to stretch it further by another $5.0 \mathrm{~cm}$ ?

Sunita Kumari

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 $110 \mathrm{~N} / \mathrm{m}$. 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?

Sunita Kumari

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 7-28 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) $x=-5.0 \mathrm{~cm} ?$

Sunita Kumari

Numerade Educator

Problem 30

$$
\begin{aligned}
&\text { In Fig. 7-10a, a block of mass } m \\
&\text { lies on a horizontal frictionless sur- } \\
&\text { face and is attached to one end of a } \\
&\text { horizontal spring (spring constant } \\
&k) \text { whose other end is fixed. The } \\
&\text { block is initially at rest at the posi- } \\
&\text { tion where the spring is unstretched } \\
&(x=0) \text { when a constant horizontal } \\
&\text { force } \vec{F} \text { in the positive direction of the } x \text { axis is applied to it. A plot } \\
&\text { of the resulting kinetic energy of the block versus its position } x \text { is } \\
&\text { shown in Fig. 7-29. The scale of the vertical axis is set by } K_{s}=6.0 \mathrm{~J} \text {. } \\
&\text { (a) What is the magnitude of } \vec{F} ? \text { (b) What is the value of } k \text { ? }
\end{aligned}
$$

Ze-Han Lee

Numerade Educator

Problem 31

As a $2.5 \mathrm{~kg}$ body moves in the positive direction along an $x$ axis, a single force acts on it. The force is given by $F_{x}=-6 x \mathrm{~N}$, with $x$ in meters. The velocity at $x=3.5 \mathrm{~m}$ is $8.5 \mathrm{~m} / \mathrm{s}$. (a) Find the velocity of the body at $x=4.5 \mathrm{~m}$. (b) Find the positive value of $x$ at which the body has a velocity of $5.5 \mathrm{~m} / \mathrm{s}$.

Sunita Kumari

Numerade Educator

Problem 32

Figure $7-30$ 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 $x=12 \mathrm{~cm}$. 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) $x=+5.0$
from $x_{i}=+8.0 \mathrm{~cm}$ to (a) $x=+50-\quad-I_{s} \mid$
$\mathrm{cm}$, (b) $x=-5.0 \mathrm{~cm}$, (c) $x=-8.0 \quad$ Figure 7-30 Problem 32 .
$\mathrm{cm}$, and (d) $x=-10.0 \mathrm{~cm}$ ?

Sunita Kumari

Numerade Educator

Problem 33

The block in Fig. 7-10a 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?

Sunita Kumari

Numerade Educator

Problem 34

$$
\begin{aligned}
& \text { A } 15 \mathrm{~kg} \text { brick moves along an } \\
&x \text { axis. Its acceleration as a function } \\
&\text { of its position is shown in Fig. } 7-31 \text {. } \\
&\text { The scale of the figure's vertical } \\
&\text { axis is set by } a_{s}=24 \mathrm{~m} / \mathrm{s}^{2} \text {. What is } \\
&\text { the net work performed on the } \\
&\text { brick by the force causing the accel- } \\
&\text { eration as the brick moves from } \\
&x=0 \text { to } x=8.0 \mathrm{~m} ?
\end{aligned}
$$

Sunita Kumari

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 $2.5 \mathrm{~kg}$ block moves in a straight line on a horizontal friction-
less surface under the influence of a
force that varies with position as
shown in Fig. 7-32. 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 $x=8.0 \mathrm{~m}$ ?
Figure 7-32 Problem 36.

Sunita Kumari

Numerade Educator

Problem 37

Figure 7-33 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) $x=4.0 \mathrm{~m}$, (b) $x=7.0 \mathrm{~m}$, and (c) $x=9.0 \mathrm{~m}$ ? What is the particle's speed and direction of travel when it reaches (d) $x=4.0 \mathrm{~m}$, (e) $x=7.0 \mathrm{~m}$, and (f) $x=9.0 \mathrm{~m}$ ?
Figure 7-33 Problem $37 .$

Sunita Kumari

Numerade Educator

Problem 38

A $1.0 \mathrm{~kg}$ 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 $x=2.0 \mathrm{~m} ?$

Sunita Kumari

Numerade Educator

Problem 39

A particle of mass $0.020 \mathrm{~kg}$ moves along a curve with velocity $5.0 \hat{\mathrm{i}}+18 \hat{\mathrm{k}} \mathrm{m} / \mathrm{s}$. After some time, the velocity changes to $9.0 \hat{\mathrm{i}}+22 \hat{\mathrm{j}}$ $\mathrm{m} / \mathrm{s}$ due to the action of a single force. Find the work done on the particle during this interval of time.

Sunita Kumari

Numerade Educator

Problem 40

A can of sardines is made to move along an $x$ axis from $x=0.25 \mathrm{~m}$ to $x=2.25 \mathrm{~m}$ 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?

Sunita Kumari

Numerade Educator

Problem 41

Only one force is acting on a $2.8 \mathrm{~kg}$ particle-like object whose position is given by $x=4.0 t-5.0 t^{2}+2.0 t^{3}$, with $x$ in meters and $t$ in seconds. What is the work done by the force from $t=0 \mathrm{~s}$ to $t=6.0 \mathrm{~s}$ ?

Sunita Kumari

Numerade Educator

Problem 42

Figure 7-34 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 $h=1.25 \mathrm{~m}$, 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 $28.0 \mathrm{~N}$. What is the change in the kinetic energy of the cart during the move?

Sunita Kumari

Numerade Educator

Problem 43

A force of 5.0 N acts on a 15 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 $880 \mathrm{~J}$ of work on the skier as the skier moves a distance of $7.0$ $\mathrm{m}$ 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) $2.0 \mathrm{~m} / \mathrm{s}$ ?

Sunita Kumari

Numerade Educator

Problem 45

Across a horizontal floor, a $102 \mathrm{~kg}$ block is pulled at a constant speed of $5.5 \mathrm{~m} / \mathrm{s}$ by an applied force of $125 \mathrm{~N}$ directed $38^{\circ}$ above the horizontal. Calculate the rate at which the force does work on the block.

Sunita Kumari

Numerade Educator

Problem 46

The loaded cab of an elevator has a mass of $5.0 \times 10^{3} \mathrm{~kg}$ 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?

Sunita Kumari

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 position 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.

Eduard Sanchez

Numerade Educator

Problem 48

A $0.35 \mathrm{~kg}$ ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal spring $(k=450 \mathrm{~N} / \mathrm{m})$ 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?

Sunita Kumari

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 $3.0 \mathrm{~min}$, 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 velocity 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 $-15 \mathrm{~W}$, what is the velocity of the object?

Sunita Kumari

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