Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 5

Force and Motion-1 - all with Video Answers

Educators

Chapter Questions

Problem 1

When two perpendicular forces $9.0 \mathrm{~N}$ (toward positive $x$ ) and $7.0 \mathrm{~N}$ (toward positive $y$ ) act on a body of mass $6.0 \mathrm{~kg}$, what are the (a) magnitude and (b) direction of the acceleration of the body?

Krystal K

Krystal K

Numerade Educator

Problem 2

Two horizontal forces act on a $2.5 \mathrm{~kg}$ chopping block that can slide over a frictionless kitchen counter, which lies in an $x y$ plane. One force is $\vec{F}_{1}=(3.0 \mathrm{~N}) \hat{1}+(4.0 \mathrm{~N}) \hat{\mathrm{j}}$. Find the acceleration of the chopping block in unit-vector notation when the other force is (a) $\vec{F}_{2}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}+(-4.0 \mathrm{~N}) \hat{\mathrm{j}}_{,}$(b) $\vec{F}_{2}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}+(4.0 \mathrm{~N}) \hat{\mathrm{j}}$, and (c) $\vec{F}_{2}^{*}=(3.0 \mathrm{~N}) \hat{\mathrm{i}}+(-4.0 \mathrm{~N}) \hat{\mathrm{j}}$.

Arpit Gupta

Numerade Educator

Problem 3

A body has an acceleration of $3.00 \mathrm{~m} / \mathrm{s}^{2}$ at $30.0^{\circ}$ to the positive direction of an $x$ axis. The mass of the body is $2.00 \mathrm{~kg}$. Find (a) the $x$ component and (b) the $y$ component of the net force acting on the body. (c) What is the net force in unit-vector notation?

Arpit Gupta

Numerade Educator

Problem 4

A particle is to move along a line at the constant velocity $\vec{v}=(2 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}-(3 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$. During the motion of the particle, we assume that two forces are acting on it. If one of the forces is $\vec{F}=(2 \mathrm{~N}) \hat{\mathrm{i}}+(-5 \mathrm{~N}) \hat{\mathrm{j}}$, find the other force.

Arpit Gupta

Numerade Educator

Problem 5

Three astronauts, propelled by jet backpacks, push and guide a 120 $\mathrm{~kg}$ asteroid toward a processing dock, exerting the forces shown in Fig. $5-19$, with $F_{1}=32 \mathrm{~N}, F_{2}=55 \mathrm{~N}$, $F_{3}=41 \mathrm{~N}, \theta_{1}=30^{\circ}$, and $\theta_{3}=60^{\circ} .$ What is the asteroid's acceleration
(a) in unit-vector notation and as
(b) a magnitude and (c) a direction relative to the positive direction of the $x$ axis?
Figure 5-19 Problem $5 .$

Arpit Gupta

Numerade Educator

Problem 6

In a two-dimensional tug-of-war, Alex, Betty, and Charles pull horizontally on an automobile tire at the angles shown in the overhead view of Fig. $5-20$. The tire remains stationary in spite of the three pulls. Alex pulls with force $\vec{F}_{A}$ of magnitude $250 \mathrm{~N}$, and Charles pulls with force $\vec{F}_{C}$ of magnitude $170 \mathrm{~N}$. Note that the direction of $\vec{F}_{C}$ is not given. What is the magnitude of Betty's force $\vec{F}_{B}$ ?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 7

There are two forces on the $2.00$ $\mathrm{kg}$ box in the overhead view of Figure 5-20 Problem $6 .$ Fig. $5-21$, but only one is shown. For $F_{1}=20.0 \mathrm{~N}, a=12.0 \mathrm{~m} / \mathrm{s}^{2}$, and $\theta=30.0^{\circ}$, find the second force (a) in unit-vector notation and as (b) a magnitude and (c) an angle relative magnitude and $(\mathrm{c})$ an angle relative to the positive direction of the $x$ axis.

Arpit Gupta

Numerade Educator

Problem 8

A $1.50 \mathrm{~kg}$ object is subjected to three forces that give it an acceleration $\vec{a}=-\left(8.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. If two of the three forces are $\vec{F}_{1}=(30.0 \mathrm{~N}) \hat{\mathrm{i}}+(16.0 \mathrm{~N}) \hat{\mathrm{j}}$ and $\vec{F}_{2}=-(12.0 \mathrm{~N}) \overrightarrow{\mathrm{i}}+(8.00 \mathrm{~N}) \hat{\mathrm{j}}$, find the third force.

Arpit Gupta

Numerade Educator

Problem 9

In an $x y$ plane, a $0.450 \mathrm{~kg}$ object moves in such a way that $x(t)=$ $-16.0+3.00 t-5.00 t^{3}$ and $y(t)=26.0+8.00 t-10.0 t^{2}$, where $x$ and $y$ are measured in meters and $t$ in seconds. At $t=0.800 \mathrm{~s}$, find (a) the magnitude and (b) the angle, relative to the positive direc- tion of the $x$ axis, of the net force on the object, and (c) the angle of the object's travel direction.

Arpit Gupta

Numerade Educator

Problem 10

A $0.150 \mathrm{~kg}$ particle moves along an $x$ axis according to $x(t)=-13.00+2.00 t+4.00 t^{2}-3.00 t^{3}$, with $x$ in meters and $t$ in seconds. In unit-vector notation, what is the net force acting on the particle at $t=2.60 \mathrm{~s}$ ?

Arpit Gupta

Numerade Educator

Problem 11

A $3.0 \mathrm{~kg}$ object is driven along an $x$ axis by a variable force that is directed along that axis. Its position is given by $x=4.0 \mathrm{~m}+$ $(5.0 \mathrm{~m} / \mathrm{s}) t+k t^{2}-\left(3.0 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}$, where $x$ is measured in meters and $t$ in seconds. The factor $k$ is a constant. At $t=4.0 \mathrm{~s}$, the force on the particle has a magnitude of $37 \mathrm{~N}$ and is in the negative direction of the axis. Find the value of $k$.

Arpit Gupta

Numerade Educator

Problem 12

Two horizontal forces $\vec{F}_{1}$ and $\vec{F}_{2}$ act on a $4.0 \mathrm{~kg}$ disk that slides over frictionless ice, on which an $x y$ coordinate system is laid out. Force $\vec{F}_{1}$ is in the positive direction of the $x$ axis and has a magnitude of $7.0 \mathrm{~N}$. Force $\vec{F}_{2}$ has a magnitude of $9.0$ $\mathrm{~N}$. Figure $5-22$ gives the $x$ component $v_{x}$ of the velocity of the $\quad$ Figure $5-22$ Problem 12 . disk as a function of time $t$ during the sliding. What is the angle between the constant directions of forces $\vec{F}_{1}$ and $\vec{F}_{2}$ ?

Arpit Gupta

Numerade Educator

Problem 13

Two particles of masses $m$ and $2 m$ are placed on a smooth horizontal table. A string, which joins these two masses, hangs over the edge supporting a pulley, which suspends a particle of mass $3 m$, as shown in Fig. $5-23$. The pulley has negligible mass. The two parts of the string on the table are parallel and perpendicular to the edge of the table. The hanging parts of the string are vertical. Find the acceleration of the particle of mass $3 \mathrm{~m}$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 14

A block with a weight of $4.0 \mathrm{~N}$ is at rest on a horizontal surface. A $1.0 \mathrm{~N}$ upward force is applied to the block by means of an attached vertical string. What are the (a) magnitude and (b) direction of the force of the block on the horizontal surface?

Arpit Gupta

Numerade Educator

Problem 15

(a) An $11.0 \mathrm{~kg}$ salami is supported by a cord that runs to a spring scale, which is supported by a cord hung from the ceiling (Fig. $5-24 a$ ). What is the reading on the scale, which is marked in SI weight units? (This is a way to measure weight by a deli owner.) (b) In Fig. $5-24 b$ the salami is supported by a cord that runs around a pulley and to a scale. The opposite end of the scale is attached by a cord to a wall.
What is the reading on the scale? (This is the way by a physics major.)
(c) In Fig. $5-24 c$ the wall has been replaced with a second $11.0 \mathrm{~kg}$ salami, and the assembly is stationary. What is the reading on the scale? (This is the way by a deli owner who was once a physics major.)
Figure 5-24 Problem $15 .$

Arpit Gupta

Numerade Educator

Problem 16

Some insects can walk below a thin rod (such as a twig) by hanging from it. Suppose that such an insect joint has mass $m$ and hangs from a horihas mass $m$ and hangs from a horizontal rod as shown in Fig. $5-25$, with angle $\theta=40^{\circ}$. Its six legs are all untions nearest the body are horizontal. (a) What is the ratio of the tension in each tibia (forepart of a leg) to the insect's weight? (b) If the insect straightens out its legs somewhat, does the tension in each tibia increase, decrease, or stay the same?

Arpit Gupta

Numerade Educator

Problem 17

In Fig. $5-26$, let the mass of the block be $8.5 \mathrm{~kg}$ and the angle $\theta$ be $30^{\circ}$. Find (a) the tension in the cord and (b) the normal force acting on the block. (c) If the cord is cut, find the magnitude of the resulting acceleration of the block.

Arpit Gupta

Numerade Educator

Problem 18

In April 1974, John Massis
Figure 5-25 Problem $16 .$ of Belgium managed to move two passenger railroad cars. He did so by clamping his teeth down on a bit that was attached to the cars with a rope and then ing backward while pressing his feet against the railway ties. The together weighed $700 \mathrm{kN}$ (about 80 tons). Assume that he pulled a constant force that was $2.5$ times his body weight, at an upward gle $\theta$ of $30^{\circ}$ from the horizontal. His mass was $80 \mathrm{~kg}$, and he move 17 . the cars by $1.0 \mathrm{~m}$. Neglecting any retarding force from the wheel rote tion, find the speed of the cars at the end of the pull.

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Problem 19

A $550 \mathrm{~kg}$ rocket sled can be accelerated at a constant rate from rest to $1650 \mathrm{~km} / \mathrm{h}$ in $2.0 \mathrm{~s}$. What is the magnitude of the required net force?

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Problem 20

A car traveling at $63 \mathrm{~km} / \mathrm{h}$ hits a bridge abutment. A passenger in the car moves forward a distance of $65 \mathrm{~cm}$ (with respect to the road) while being brought to rest by an inflated air bag. What magnitude of force (assumed constant) acts on the passenger's upper torso, which has a mass of $41 \mathrm{~kg}$ ?

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Problem 21

A constant horizontal force $\vec{F}_{\text {a }}$ pushes a $2.00 \mathrm{~kg}$ FedEx package across a frictionless floor on which an $x y$ coordinate system has
$$
v_{x}(\mathrm{~m} / \mathrm{s})
$$
Figure 5-27 Problem 21.
been drawn. Figure 5-27 gives the package's $x$ and $y$ velocity
components versus time $t$. What are the (a) magnitude and
(b) direction of $\vec{F}_{u}$ ?

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Problem 22

A customer sits in an amusement park ride in which the com-
partment is to be pulled downward in the negative direction of a $y$
axis with an acceleration magnitude of $1.24 g$, with $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$. A
$0.567 \mathrm{~g}$ coin rests on the customer's knee. Once the motion begins
and in unit-vector notation, what is the coin's acceleration relative
to (a) the ground and (b) the customer? (c) How long does
the coin take to reach the compartment ceiling, $2.20 \mathrm{~m}$ above the
knee? In unit-vector notation, what are (d) the actual force on
the coin and (e) the apparent force according to the customer's
measure of the coin's acceleration?

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Problem 23

Tarzan, who weighs $860 \mathrm{~N}$, swings from a cliff at the end of a $20.0 \mathrm{~m}$ vine that hangs from a high tree limb and initially makes an angle of $22.0^{\circ}$ with the vertical. Assume that an $x$ axis extends horizontally away from the cliff edge and a $y$ axis extends upward. Immediately after Tarzan steps off the cliff, the tension in the vine is $760 \mathrm{~N}$. Just then, what are (a) the force on him from the vine in unit-vector notation and the net force on him (b) in unit-vector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the $x$ axis? What are the (e) magnitude and (f) angle of Tarzan's acceleration just then?

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Problem 24

There are two horizontal forces on the $2.0 \mathrm{~kg}$ box in the overhead view of Fig. $5-28$ but only one (of magnitude $F_{1}=30 \mathrm{~N}$ ) is shown. The
reaterhead

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Problem 25

Sunjamming. A "sun yacht" is a spacecraft with a large sail that is pushed by sunlight. Although such a push is tiny in everyday cir-
cumstances, it can be large enough to send the spacecraft outward
from the Sun on a cost-free but slow trip. Suppose that the spacecraft
has a mass of $900 \mathrm{~kg}$ and receives a push of $20 \mathrm{~N}$. (a) What is the mag-
nitude of the resulting acceleration? If the craft starts from rest, (b)
how far will it travel in 1 day and (c) how fast will it then be moving?

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Problem 26

The tension at which a fishing line snaps is commonly called
the line's "strength." What minimum strength is needed for a line
that is to stop a salmon of weight $90 \mathrm{~N}$ in $11 \mathrm{~cm}$ if the fish is initially
drifting at $2.8 \mathrm{~m} / \mathrm{s}$ ? Assume a constant deceleration.

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Problem 27

A subatomic particle moves horizontally, with a speed of
$1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}$, into a region where a uniform vertical electric force
of $5.5 \times 10^{-16} \mathrm{~N}$ acts on it. Assuming the subatomic particle is an
electron (the mass of the electron is $9.11 \times 10^{-31} \mathrm{~kg}$ ), find the
vertical distance the particle is deflected during the time it has
moved $35 \mathrm{~mm}$ horizontally.

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Problem 28

A car that weighs $1.30 \times 10^{4} \mathrm{~N}$ is initially moving at $35 \mathrm{~km} / \mathrm{h}$
when the brakes are applied and the car is brought to a stop in $15 \mathrm{~m}$.
Assuming the force that stops the car is constant, find (a) the
magnitude of that force and (b) the time required for the change in
speed. If the initial speed is doubled, and the car experiences the
same force during the braking, by what factors are (c) the stopping
distance and (d) the stopping time multiplied? (There could be a
lesson here about the danger of driving at high speeds.)

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Problem 29

A firefighter who weighs $689 \mathrm{~N}$ slides down a vertical pole
with an acceleration of $2.00 \mathrm{~m} / \mathrm{s}^{2}$, directed downward. What are the
(a) magnitude and (b) direction (up or down) of the vertical force
on the firefighter from the pole and the (c) magnitude and (d) di-
rection of the vertical force on the pole from the firefighter?

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Problem 30

The high-speed winds around a tornado can drive projectiles into trees, building walls, and even metal traffic signs. In a laboratory simulation, a standard wood toothpick was shot by pneumatic gun into an oak branch. The toothpick's mass was $0.13 \mathrm{~g}$, its speed before entering the branch was $205 \mathrm{~m} / \mathrm{s}$, and its penetration depth was $15 \mathrm{~mm}$. If its speed was decreased at a uniform rate, what was the magnitude of the force of the branch on the toothpick?

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Problem 31

A block is sent moving up a long, inclined, frictionless plane with an initial speed $v_{0}=2.77 \mathrm{~m} / \mathrm{s}$. Assume that the angle of inclination is $\theta=28.0^{\circ}$. Find (a) the distance in which the block moves to its highest point, (b) the time the block takes to get there, and (c) its speed when it gets back to the bottom. $\theta_{2}=30.0^{\circ}$. In unit-vector notation, what is the third force if the lemon Figure 5-29 Problem

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Problem 32

half (a) is stationary, (b) has the con-
stant velocity $\vec{v}=(13.0 \hat{i}-14.0 \hat{j}) \mathrm{m} / \mathrm{s}$, and (c) has the varying velocity $\vec{v}=(13.0 t \hat{i}-14.0 t \hat{j}) \mathrm{m} / \mathrm{s}^{2}$, where $t$ is time?

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Problem 33

A. $1500 \mathrm{~kg}$ cable car moves vertically by means of a cable that connects the ground and the top of a hill. What is the tension in the supporting cable when the cab, originally moving downward at a speed of $9.0 \mathrm{~m} /$ s , is brought to rest with constant acceleration in a distance of $38

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Problem 34

In Fig. $5-30$, a crate of mass $m=115 \mathrm{~kg}$ is pushed at constant speed up a frictionless ramp $\left(\theta=30.0^{\circ}\right)$ by a horizontal force $\vec{F}$. What are the magnitudes of (a) $\vec{F}$ and (b) the force on the crate from the ramp?

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Problem 35

An object weighs $2.50 \mathrm{~kg}$. In Figure 5-30 Problem $34 .$ time $t$, measured in seconds, the ve-
locity of the object is given by $\vec{v}=\left(7.00 r \hat{i}+2.00 r^{2} \hat{j}\right) \mathrm{m} / \mathrm{s}$. At the instant the net force on the object has a magnitude of $35.0 \mathrm{~N}$, what are (a) the direction of the net force and (b) the object's direction of travel?

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Problem 36

Holding on to a towrope moving parallel to a frictionless ski slope, a $45 \mathrm{~kg}$ skier is pulled up the slope, which is at an angle of $8.0^{\circ}$ with the horizontal. What is the magnitude $F_{\text {rope }}$ of the force on the skier from the rope when (a) the magnitude $v$ of the skier's velocity is constant at $2.0 \mathrm{~m} / \mathrm{s}$ and $(\mathrm{b}) v=2.0 \mathrm{~m} / \mathrm{s}$ as $v$ increases at a rate of $0.10 \mathrm{~m} / \mathrm{s}^{2}$ ?

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Problem 37

A boy with a mass of $35 \mathrm{~kg}$ and a sled with a mass of $6.5 \mathrm{~kg}$ are on the frictionless ice of a frozen lake, $12 \mathrm{~m}$ apart but connected by a rope of negligible mass. The boy exerts a horizontal $4.2 \mathrm{~N}$ force on the rope. What are the acceleration magnitudes of (a) the sled and (b) the boy? (c) How far from the boy's initial position do they meet?

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Problem 38

A $50 \mathrm{~kg}$ skier skis directly down a frictionless slope angled at $10^{\circ}$ to the horizontal Assume the skier moves in the negative direction of an $x$ axis along the slope. A wind force with component $F_{x}$ acts on the skier. What is $F_{x}$ if the magnitude of the skier's velocity is (a) constant, (b) increasing at a rate of $1.0 \mathrm{~m} / \mathrm{s}^{2}$, and (c) increasing at a rate of $2.0 \mathrm{~m} / \mathrm{s}^{2}$ ?

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Problem 39

A bead of mass $2.5 \times 10^{-4} \mathrm{~kg}$ is suspended from a cord. A steady horizontal breeze pushes the bead so that the cord makes a constant angle of $40^{\circ}$ with the vertical. Find (a) the tension in the cord and (b) the push magnitude.

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Problem 40

A dated box of dates of $v_{\gamma}(\mathrm{m} / \mathrm{s})$
mass $4.50 \mathrm{~kg}$, is sent sliding up a frictionless ramp at an angle of $\theta$ to the horizontal. tion of time $t$, the component $v_{x}$ of the box's velocity along $v_{x}$ of the box's velocity along up the ramp. What is the mag-
nitude of the normal force on the box from the ramp? $\quad$ Figure 5-31 Problem $40 .$

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Problem 41

Joyce needs to lower a bundle of scrap material that weighs $450 \mathrm{~N}$ from a point $6.2 \mathrm{~m}$ above the ground. For doing this exercise, Joyce uses a rope that will break if the tension in it exceeds $390 \mathrm{~N}$. Clearly if she hangs the bundle on the rope, it will break and therefore Joyce allows the bundle to accelerate downward. (a) What magnitude of the bundle's acceleration will put the rope on the verge of snapping? (b) At that acceleration, with what speed would the bundle hit the ground?

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Problem 42

In earlier days, horses pulled barges down canals in the manner shown in Fig. 5-32. Suppose the horse pulls on the rope with a force of $8600 \mathrm{~N}$ at an angle of $\theta=18^{\circ}$ to the direction of motion of the barge, which is headed straight along the positive direction of an $x$ axis. The mass of the barge is $9500 \mathrm{~kg}$, and the magnitude of its acceleration is $0.12 \mathrm{~m} / \mathrm{s}^{2}$. What are the (a) magnitude and (b) direction (relative to positive $x$ ) of the force on the barge from the water? Find 5 -32 Problem 42 .

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Problem 43

In Fig. $5-33$, a chain consisting of five links, cach of mass $0.100 \mathrm{~kg}$, is lifted vertically with constant acceleration of magnitude $a=2.50 \mathrm{~m} / \mathrm{s}^{2}$. Find the magnitudes of (a) the force on link 1 from link $2,(b)$ the force on link 2 from link $3,(c)$ the force on link 3 from link 4 , and (d) the force on link 4 from link 5 . Then find the magnitudes of (e) the force $\vec{F}$ on the top link from the person lifting the chain and (f) the net force accelerating each link.

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Problem 44

A lamp hangs vertically from a cord in a de- scending elevator that decelerates at $2.4 \mathrm{~m} / \mathrm{s}^{2}$. (a) If the tension in the cord is $93 \mathrm{~N}$, what is the lamp's mass? (b) What is the cord's tension Problem $43 .$ when the elevator ascends with an upward acceleration of $2.4 \mathrm{~m} / \mathrm{s}^{2}$ ?

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Problem 45

An elevator cab that weighs $29.0 \mathrm{kN}$ moves upward. What is the tension in the cable if the cab's speed is (a) increasing at a rate of $1.50 \mathrm{~m} / \mathrm{s}^{2}$ and (b) decreasing at a rate of $1.50 \mathrm{~m} / \mathrm{s}^{2}$ ?

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Problem 46

An elevator cab is pulled upward by a cable. The cab and its single occupant have a combined mass of $2000 \mathrm{~kg}$. When that occupant drops a coin, its acceleration relative to the cab is $8.30 \mathrm{~m} / \mathrm{s}^{2}$ downward. What is the tension in the cable?

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Problem 47

The Zacchini family was renowned for their human-cannonball act in which a family member was shot from a cannon using either elastic bands or compressed air. In one version of the act, Emanuel Zacchini was shot over three Ferris wheels to land in a net at the same height as the open end of the cannon and at a range of $69 \mathrm{~m}$. He was propelled inside the barrel for $5.2 \mathrm{~m}$ and launched at an angle of $53^{\circ}$. If his mass was $85 \mathrm{~kg}$ and he underwent constant acceleration inside the barrel, what was the magnitude of the force propelling him? (Hint: Treat the launch as though it were along a ramp at $53^{\circ}$. Neglect air drag.)

Averell Hause

Carnegie Mellon University

Problem 48

In Fig. $5-34$, elevator cabs $A$ and $B$ are connected by a short cable and can be pulled upward or lowered by the cable above cab $A$. Cab $A$ has mass $1700 \mathrm{~kg}$; Figure 5-34 cab $B$ has mass $1200 \mathrm{~kg}$. A $12.0 \mathrm{~kg}$ box of catnip lies Problem 48 . on the floor of cab $A$. The tension in the cable con- necting the cabs is $1.91 \times 10^{4} \mathrm{~N}$. What is the magnitude of the normal necting the cabs is $1.91 \times 10^{4} \mathrm{~N}$. force on the box from the floor?

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Problem 49

In Fig. $5-35$, a block of mass $m=5.00 \mathrm{~kg}$ is pulled along a horizontal frictionless floor by a cord that exerts a force of magnitude $F=120 \mathrm{~N}$ at an angle $\theta=25.0^{\circ}$. (a) What is the magnitude of the block's acceleration? (b) The force magnitude $F$ is slowly increased. What is its value just before the block is lifted (completely) off the floor? (c) What is the magnitude of the block's acceleration just before it is lifted (completely) off the floor?

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Problem 50

In Fig 5-36, three ballot boxes are
connected by corcts, one of which wraps over a pulley having negligible friction on its axie and negligible mass. The three masses are $m_{A}=30.0 \mathrm{~kg}, m_{B}$ $=30.0 \mathrm{~kg}$, and $m_{C}=10.0 \mathrm{~kg}$. When
the nssembly is relenced from rest, (a) Figure 5-36 Problem 50 the assembly is releascd from rest, (a) necting $B$ and $C$ and $(b)$ how far does $A$ necting $B$ and $C$ and (b) how far does $A$ move in the first $0.250 \mathrm{~s}$ (assuming it does not rench the pulley) 2 first $0.250 \mathrm{~s}$ (assuming it does not reach the pulley)?

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Problem 51

Figure 5-37 shows two blocks connected by a cord (of negligible mass) that passes over a frictionless pulley (also of negligible mass). The arrangement is known as Atwood's machine. One block has mass $m_{1}=1_{3} 30 \mathrm{~kg}^{-}$the other has mass $m_{2}=2.80 \mathrm{~kg}$. What are (a) the magnitude of the blocks acceleration and (b) the tension in the cord?

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Problem 52

A $93 \mathrm{~kg}$ man lowers himself to the ground from a height of $10.0 \mathrm{~m}$ by holding onto a rope that runs over a frictionless pulley to a $65 \mathrm{~kg}$ sandbag. With what speed does the man hit the ground if he Problems 51 and $65 .$ started from rest?

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Problem 53

As shown in Fig. 5-38, body $C$ (2.9
$\mathrm{kg})$ and body $D(1.9 \mathrm{~kg})$ are suspended from a rigid support by inextensible wires $B$ and $A$, each of length $1.0 \mathrm{~m}$. Wire $B$ has negligible mass; wire $A$ has a uniform density of $0.20 \mathrm{~kg} / \mathrm{m}$. The whole system undergoes an upward acceleration of magnitude $0.50 \mathrm{~m} / \mathrm{s}^{2}$. Find the tension at the midpoint in (a) wire $A$ and (b) wire $B$.
Figure 5-38 Problem 53

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Problem 54

Figure 5-39 shows four penguins that are being playfully pulled along very slippery (frictionless) ice by a curator. The masses of three penguins and the tension in two of the cords are $m_{1}=12 \mathrm{~kg}$, $m_{3}=15 \mathrm{~kg}_{n} m_{4}=20 \mathrm{~kg}, T_{2}=111 \mathrm{~N}$, and $T_{4}=222 \mathrm{~N} .$ Find the penguin mass $m_{2}$ that is not given.
Figure 5-39 Problem $54 .$

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Problem 55

Two blocks are in contact on a frictionless table. A horizontal force is applied to the larger block, as shown in Fig. $5-40$. (a) If $m_{1}=2.3 \mathrm{~kg}, m_{2}=1.2 \mathrm{~kg}$, and $F$ $=3.2 \mathrm{~N}$, find the magnitude of the force between the two blocks. (b) Show that if a force of the same magnitude $F$ is applied to the smaller block but in the opposite di- rection, the magnitude of the force between Problem $55 .$
the blocks is $2.1 \mathrm{~N}$, which (c) Explain the difference.

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Problem 56

In Fig. $5-41 a$, a constant horizontal force $\vec{F}_{\mathrm{a}}$ is applied to block $A$, which pushes against block $B$ with a $15.0 \mathrm{~N}$ force directed horizontally to the right. In Fig. $5-41 b$, the same force $\vec{F}_{a}$ is applied to block $B$; now block $A$ pushes on block $B$ with a $10.0 \mathrm{~N}$ force directed horizontally to the left. The blocks have a combined mass of $12.0 \mathrm{~kg}$. What are the magnitudes of (a) their acceleration in Fig. $5-41 a$ and $(b)$ force $\vec{F}_{a}$ ?
Figure 5-41 Problem $56 .$

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Problem 57

Two blocks of masses $m_{1}=8.0 \mathrm{~kg}$ and $m_{2}=4.0 \mathrm{~kg}$ are connected by a string as shown in Fig. 5-42, over a frictionless pulley of negligible mass. What is the acceleration magnitude of the two-block system if $\theta=30.0^{\circ}$ ?

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Problem 58

Figure $5-43$ shows a man sitting in a bosun's chair that dangles from a massless rope, which runs over a massless, frictionless pulley and back down to the man's hand. The combined mass of man and chair is $103.0 \mathrm{~kg}$. With what force magnitude must the man pull on the rope if he is to rise (a) with a constant velocity and (b) with an upward acceleration of $1.30 \mathrm{~m} / \mathrm{s}^{2}$ ? (Hint: A free-body diagram can really help.) If the rope on the right extends to the ground and is pulled by a co-worker, with what force magnitude must the co-worker pull Figure 5-43 Problem 58 .
for the man to rise (c) with a constant velocity and (d) with an upward acceleration of $1.30 \mathrm{~m} / \mathrm{s}^{2} ?$ What is the magnitude of the force on the ceiling from the pulley system in (c) part a, (f) part b, (g) part $\mathrm{c}$, and $(\mathrm{h})$ part $\mathrm{d}$ ?

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Problem 59

A $10 \mathrm{~kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a $15 \mathrm{~kg}$ package on the ground (Fig. 5-44). (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops has becn lifted, the monkey stops its climb and holds on to the rope, what are the (b) magnitude and (c) what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope?

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Problem 60

Figure $5-35$ shows a $5.00 \mathrm{~kg}$ block being pulled along a frictionless floor by a cord that applies a force of constant magnitude $15.0 \mathrm{~N}$ but with an angle $\theta(t)$ that varies with time. When angle $\theta=$ $25.0^{0}$, at what rate is the acceleration of the block changing if (a) $\theta(t)=\left(2.00 \times 10^{-2} \mathrm{deg} / \mathrm{s}\right) t$ and $(\mathrm{b}) \theta(t)=-\left(2.00 \times 10^{-2} \mathrm{deg} / \mathrm{s}\right) t ?$ (Hint: The angle should be in radians)

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 61

A hot-air balloon of mass $M$ is descending vertically with downward acceleration of magnitude $a$. How much mass (ballast) must be thrown out to give the balloon an upward acceleration of magnitude $a$ ? Assume that the upward force from the air (the lift) does not change because of the decrease in mass.

Averell Hause

Carnegie Mellon University

Problem 62

In shot putting, many athletes elect to launch the shot at an angle that is smaller than the theoretical one (about $42^{\circ}$ ) at which the distance of a projected ball at the same speed and height is greatest. One reason has to do with the speed the athlete can give the shot during the acceleration phase of the throw. Assume that a $7.260 \mathrm{~kg}$ shot is accelerated along a straight path of length $1.650 \mathrm{~m}$ by a constant applied force of magnitude $380.0 \mathrm{~N}$, starting with an initial speed of $2.500 \mathrm{~m} / \mathrm{s}$ (due to the athlete's preliminary motion). What is the shot's speed at the end of the acceleration phase if the angle between the path and the horizontal is (a) $30.00^{\circ}$ and (b) $42.00^{\circ}$ ? (Hint: Treat the motion as though it were along a ramp at the given angle.) (c) By what percentage is the launch speed decreased if the athlete increases the angle from $30.00^{\circ}$ to $42.00^{\circ}$ ?

Averell Hause

Carnegie Mellon University

Problem 63

Figure $5-45$ gives, as a function of time $t$, the force component $F_{x}$ that acts on a $3.00 \mathrm{~kg}$ ice block that can move only along the $x$ axis. At $t=0$, the block is moving in the positive direction of the axis, with a speed of $3.0 \mathrm{~m} / \mathrm{s}$. What are its (a) speed and (b) direction of travel at $t=11 \mathrm{~s}$ ?
$$
F_{x}(\mathrm{~N})
$$
Figure 5-45 Problem $63 .$

Neelesh Sharma

Numerade Educator

Problem 64

Figure $5-46$ shows a box of mass $m_{2}=1.0 \mathrm{~kg}$ on a frictionless plane inclined at angle $\theta=30^{\circ}$. It is connected by a cord of negligible mass to a box of mass $m_{1}=2.5 \mathrm{~kg}$ on a horizontal frictionless surface. The pulley is frictionless and massless. (a) If the magnitude of horizontal force $\vec{F}$ is $2.3 \mathrm{~N}$, what is the tension in the connecting cord? (b) What is the largest value the magnitude of $\vec{F}$ may have without the cord becoming slack?
Figure 5-46 Problem 64.

Neelesh Sharma

Numerade Educator

Problem 65

Figure 5-37 shows Atwood's machine, in which two containers are connected by a cord (of negligible mass) passing over a frictionless pulley (also of negligible mass). At time $t=0$, container 1 has mass $1.30 \mathrm{~kg}$ and container 2 has mass $2.80 \mathrm{~kg}$, but container 1 is losing mass (through a leak) at the constant rate of $0.200 \mathrm{~kg} / \mathrm{s}$. At what rate is the acceleration magnitude of the containers changing at (a) $t=0$ and (b) $t=3.00 \mathrm{~s}$ ? (c) When does the acceleration reach its maximum value?

Averell Hause

Carnegie Mellon University

Problem 66

Figure $5-47$ shows a section of a cable-car system. The maximum permissible mass of each car with occupants is $2750 \mathrm{~kg}$. The cars, riding on a support cable, are pulled by a second cable attached to the support tower on each car. Assume clined at angle $\theta=35^{\circ}$. What is the difference in tension between adjacent sections of pull cable if the cars are at the maximum permissible mass and are being accelerated up the incline at $0.81 \mathrm{~m} / \mathrm{s}^{2}$ ?

Neelesh Sharma

Numerade Educator

Problem 67

Figure 5-48 shows three blocks attached by cords that loop over frictionless pulleys Block $B$ lies on a frictionless table; the masses are $m_{d}=6.00 \mathrm{~kg}, m_{R}=8.00 \mathrm{~kg}$, and $m_{C} A$ $=10.0 \mathrm{~kg}$. When the blocks are released, what is the cord at the right?

Neelesh Sharma

Numerade Educator

Problem 68

A shot putter launches a $7.260 \mathrm{~kg}$ shot by pushing it along a straight line of length $1.650 \mathrm{~m}$ and at an angle of $34.10^{\circ}$ from the horizontal, accelerating the shot to the launch speed from its initial speed of $2.500 \mathrm{~m} / \mathrm{s}$ (which is due to the athlete's preliminary motion). The shot leaves the hand at a height of $2.110 \mathrm{~m}$ and at an angle of $34.10^{\circ}$, and it lands at a horizontal distance of $15.98 \mathrm{~m}$. What is the magnitude of the athlete's average force on the shot during the acceleration phase? (Hint: Treat the motion during the acceleration phase as though it were along a ramp at the given angle.)

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