Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 12

Equilibrium and Elasticity - all with Video Answers

Educators

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Chapter Questions

Problem 1

Because $g$ varies so little over the extent of most structures, any structure's center of gravity effectively coincides with its center of mass. Here is a fictitious example where $g$ varies more significantly. Figure $12-25$ shows an array of six particles, each with mass $m$ , fixed to the edge of a rigid structure of negligible mass. The distance between adjacent particles along the edge is 2.00 $\mathrm{m}$ . The following table gives the value of $g$ $\left(\mathrm{m} / \mathrm{s}^{2}\right)$ at each particle's location. Using the
coordinate system shown, find (a) the $x$ coordinate $x_{\text { com }}$ and $(b)$ the $y$ coordinate $y_{\text { com }}$ of the center of mass of the six-particle system. Then find (c) the $x$ coordinate $x_{\text { cog }}$ and $(\mathrm{d})$ the $y$ coordinate $y_{\text { cog }}$ of the center of gravity of the six-particle system.
$$
\begin{array}{|c|c|c|c|}\hline \text { Particle } & {g} & {\text { Particle }} & {g} \\ \hline 1 & {8.00} & {4} & {7.40} \\ \hline 2 & {7.80} & {5} & {7.60} \\ \hline 3 & {7.60} & {6} & {7.80} \\ \hline\end{array}
$$

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

An automobile with a mass of 1360 $\mathrm{kg}$ has 3.05 $\mathrm{m}$ between the front and rear axles. Its center of gravity is located 1.78 $\mathrm{m}$ behind
the front axle. With the automobile on level ground, determine the
magnitude of the force from the ground on (a) each front wheel (assuming equal forces on the front wheels) and (b) each rear wheel (assuming equal forces on the rear wheels).

Averell Hause

Carnegie Mellon University

Problem 3

In Fig. $12-26,$ a uniform sphere of mass $m=0.85 \mathrm{kg}$ and radius $r=4.2 \mathrm{cm}$ is held in place by a massless rope attached to a frictionless wall a distance $L=8.0 \mathrm{cm}$ above the center of the sphere. Find (a) the tension
in the rope and (b) the force on the sphere from the wall.

Keshav Singh

Numerade Educator

Problem 4

An archer's bow is drawn at its midpoint until the tension in the string is equal to the
force exerted by the archer. What is the angle
between the two halves of the string?

Supratim Pal

Numerade Educator

Problem 5

A rope of negligible mass is stretched horizontally between two supports that are 3.44 $\mathrm{m}$ apart. When an object of weight 3160 $\mathrm{N}$ is hung at the center of the rope, the rope is observed to sag by 35.0 $\mathrm{cm} .$ What is the tension in
the rope?

Keshav Singh

Numerade Educator

Problem 6

A scaffold of mass 60 $\mathrm{kg}$ and length 5.0 $\mathrm{m}$ is supported in a horizontal position by a vertical cable at each end. A window
washer of mass 80 kg stands at a point 1.5 $\mathrm{m}$ from one end. What is
the tension in ( a) the nearer cable and (b) the farther cable?

Averell Hause

Carnegie Mellon University

Problem 7

A 75 kg window cleaner uses a 10 kg ladder that is 5.0 $\mathrm{m}$ long. He places one end on the ground 2.5 $\mathrm{m}$ from a wall, rests the upper
end against a cracked window, and climbs the ladder. He is 3.0 $\mathrm{m}$ up
along the ladder when the window breaks. Neglect friction between
the ladder and window and assume that the base of the ladder does
not slip. When the window is on the verge of breaking, what are (a)
the magnitude of the force on the window from the ladder, (b) the
magnitude of the force on the ladder from the ground, and (c) the
angle (relative to the horizontal) of that force on the ladder?

Nicholas Mogoi

Numerade Educator

Problem 8

A physics Brady Bunch, whose weights in newtons are indicated in Fig. $12-27$ , is balanced on a seesaw. What is the number
of the person who causes the largest torque about the rotation axis
at fulcrum $f$ directed (a) out of the page and (b) into the page?

Averell Hause

Carnegie Mellon University

Problem 9

A meter stick balances horizontally on a knife-edge at the 50.0 $\mathrm{cm}$ mark. With two 5.00 $\mathrm{g}$ coins stacked over the 12.0 $\mathrm{cm}$
mark, the stick is found to balance at the 45.5 $\mathrm{cm}$ mark. What is the
mass of the meter stick?

Keshav Singh

Numerade Educator

Problem 10

The system in Fig. $12-28$ is in equilibrium, with the string in the
center exactly horizontal. Block $A$
weighs $40 \mathrm{N},$ block $B$ weighs 50 $\mathrm{N}$, and angle $\phi$ is $35^{\circ} .$ Find (a) tension
$T_{1},$ (b) tension $T_{2},$ (c) tension $T_{3},$ and
(d) angle $\theta .$

Averell Hause

Carnegie Mellon University

Problem 11

Figure $12-29$ shows a diver of weight 580 $\mathrm{N}$ standing at the
end of a diving board with a length of $L=4.5 \mathrm{m}$ and negligible mass.
The board is fixed to two pedestals
(supports) that are separated by distance $d=1.5 \mathrm{m} .$ Of the forces acting
on the board, what are the (a) magnitude and (b) direction (up or down)
of the force from the left pedestal and the (c) magnitude and (d) direction (up or down) of the force from the right pedestal? (e) Which pedestal (left or right) is being stretched, and (f) which pedestal is being compressed?

Keshav Singh

Numerade Educator

Problem 12

In Fig. $12-30$ , trying to get his car out of mud, a man ties one end of a rope around the front bumper and the other end tightly
around a utility pole 18 $\mathrm{m}$ away. He then pushes sideways on the
rope at its midpoint with a force of 550 $\mathrm{N}$ , displacing the center of
the rope $0.30 \mathrm{m},$ but the car barely moves. What is the magnitude of
the force on the car from the rope? (The rope stretches somewhat.)

Averell Hause

Carnegie Mellon University

Problem 13

Figure $12-31$ shows the anatomical structures in the
lower leg and foot that are
involved in standing on tip-toe, with the heel raised
slightly off the floor so that the foot effectively contacts
the floor only at point $P .$
Assume distance $a=5.0 \mathrm{cm}$
distance $b=15 \mathrm{cm},$ and the
person's weight $W=900 \mathrm{N} .$ Of the forces acting on the
foot, what are the (a) magnitude and (b) direction (up or down) of the force at point $A$ from the calf muscle and the (c) magnitude and (d) direction (up or
down) of the force at point $B$ from the lower leg bones?

Keshav Singh

Numerade Educator

Problem 14

In Fig. $12-32,$ a horizontal scaffold, of length 2.00 $\mathrm{m}$ and uniform mass $50.0 \mathrm{kg},$ is suspended
from a building by two cables. The
scaffold has dozens of paint cans
stacked on it at various points. The total mass of the paint cans is
75.0 $\mathrm{kg}$ . The tension in the cable at the right is 722 $\mathrm{N}$ . How far
horizontally from that cable is the center of mass of the system of
paint cans?

Averell Hause

Carnegie Mellon University

Problem 15

Forces $\vec{F}_{1}, \vec{F}_{2},$ and $\vec{F}_{3}$ act on the structure of Fig. $12-33$ shown in an overhead view. We wish to put the structure in equilibrium by applying a fourth force, at a point such as $P .$ The fourth
force has vector components $\overline{F}_{h}$ and $\vec{F}_{v}$ . We are given that $a=2.0 \mathrm{m}$, $b=3.0 \mathrm{m}, c=1.0 \mathrm{m}, F_{1}=20 \mathrm{N}, F_{2}=10 \mathrm{N},$ and $F_{3}=5.0 \mathrm{N}$ . Find $(\mathrm{a})$
$F_{h v}(\mathrm{b}) F_{v},$ and $(\mathrm{c}) d .$

Keshav Singh

Numerade Educator

Problem 16

A uniform cubical crate is 0.750 $\mathrm{m}$ on each side and weighs 500 $\mathrm{N}$ . It rests on a floor with one edge against a very small, fixed
obstruction. At what least height above the floor must a horizontal
force of magnitude 350 $\mathrm{N}$ be applied to the crate to tip it?

Averell Hause

Carnegie Mellon University

Problem 17

In Fig. $12-34,$ a uniform beam of weight 500 $\mathrm{N}$ and length 3.0 $\mathrm{m}$ is suspended horizontally. On the left it is
hinged to a wall; on the right it is supported by a cable bolted to the wall at distance $D$ above the beam. The least
tension that will snap the cable is 1200
N. (a) What value of $D$ corresponds to
that tension? (b) To prevent the cable
from snapping, should $D$ be increased
or decreased from that value?

Keshav Singh

Numerade Educator

Problem 18

In Fig. $12-35,$ horizontal scaffold $2,$ with uniform mass $m_{2}=30.0$
$\mathrm{kg}$ and length $L_{2}=2.00 \mathrm{m},$ hangs
from horizontal scaffold $1,$ with uniform mass $m_{1}=50.0 \mathrm{kg} . \mathrm{A} 20.0 \mathrm{kg}$
box of nails lies on scaffold 2, centered at distance $d=0.500 \mathrm{m}$ from the left end. What is the tension $T$ in
the cable indicated?

Averell Hause

Carnegie Mellon University

Problem 19

To crack a certain nut in a nutcracker, forces with magnitudes of at
least 40 $\mathrm{N}$ must act on its shell from
both sides. For the nutcracker of Fig. $12-36,$ with distances $L=12 \mathrm{cm}$ and
$d=2.6 \mathrm{cm},$ what are the force components $F_{\perp}$ (perpendicular to the
handles) corresponding to that 40 $\mathrm{N} ?$

Keshav Singh

Numerade Educator

Problem 20

A bowler holds a bowling ball $(M=7.2 \mathrm{kg})$ in the palm of his hand (Fig. $12-37 )$ . His upper arm is vertical; his lower arm $(1.8 \mathrm{kg})$ is
horizontal. What is the magnitude of (a) the force of the biceps
muscle on the lower arm and (b) the force between the bony
structures at the elbow contact point?

Averell Hause

Carnegie Mellon University

Problem 21

The system in Fig. $12-38$ is in equilibrium. A concrete block of
mass 225 $\mathrm{kg}$ hangs from the end of
the uniform strut of mass 45.0 $\mathrm{kg} . \mathrm{A}$
cable runs from the ground, over
the top of the strut, and down to the block, holding the block in place.
For angles $\phi=30.0^{\circ}$ and $\theta=45.0^{\circ}$
find (a) the tension $T$ in the cable
and the (b) horizontal and (c) vertical components of the force on the strut from the hinge.

Keshav Singh

Numerade Educator

Problem 22

In Fig. $12-39,$ a 55 $\mathrm{kg}$ rock climber is in a lie-back climb
along a fissure, with hands pulling on
one side of the fissure and feet
pressed against the opposite side. The fissure has width $w=0.20 \mathrm{m}$
and the center of mass of the climber
is a horizontal distance $d=0.40 \mathrm{m}$
from the fissure. The coefficient of static friction between hands and
rock is $\mu_{1}=0.40,$ and between boots
and rock it is $\mu_{2}=1.2$ . (a) What is the least horizontal pull by the hands and push by the feet that
will keep the climber stable? (b) For the horizontal pull of
(a), what must be the vertical distance $h$ between hands and feet? If the climber encounters wet rock, so
that $\mu_{1}$ and $\mu_{2}$ are reduced, what happens to $(\mathrm{c})$ the answer to $(\mathrm{a})$ and $(\mathrm{d})$
the answer to $(\mathrm{b}) ?$

Averell Hause

Carnegie Mellon University

Problem 23

In Fig. $12-40,$ one end of a uniform beam of weight 222 N is
hinged to a wall; the other end is supported by a wire that makes angles $\theta=30.0^{\circ}$ with both wall and beam.
Find (a) the tension in the wire and the
(b) horizontal and (c) vertical components of the force of the hinge on the
beam.

Surendra Kumar

Numerade Educator

Problem 24

In Fig. $12-41,$ a climber with a weight of 533.8 $\mathrm{N}$ is held by a
belay rope connected to her climbing
harness and belay device; the force of
the rope on her has a line of action through her center of mass. The indicated angles are $\theta=40.0^{\circ}$ and $\phi=$
$30.0^{\circ} .$ If her feet are on the verge of
sliding on the vertical wall, what is the coefficient of static friction between
her climbing shoes and the wall?

Averell Hause

Carnegie Mellon University

Problem 25

In Fig. $12-42,$ what magnitude of (constant) force $\vec{F}$ applied horizontally at the axle of the wheel is necessary to raise the wheel
over a step obstacle of height $h=3.00 \mathrm{cm} ?$ The wheel's radius is $r=$
$6.00 \mathrm{cm},$ and its mass is $m=0.800 \mathrm{kg} .$

Keshav Singh

Numerade Educator

Problem 26

In Fig. $12-43,$ a climber leans out against a vertical ice wall that has negligible friction. Distance $a$ is 0.914 $\mathrm{m}$ and distance $L$ is 2.10 $\mathrm{m} .$ His center of mass is distance $d=0.940 \mathrm{m}$ from the feet-ground contact point. If he is on the verge of sliding, what is the coefficient of static friction between feet and ground?

Averell Hause

Carnegie Mellon University

Problem 27

In Fig. $12-44,$ a 15 kg block is held in place via a pulley system. The
person's upper arm is vertical; the forearm is at angle $\theta=30^{\circ}$ with the horizontal. Forearm and hand together have a
mass of 2.0 $\mathrm{kg}$ , with a center of mass at distance $d_{1}=15 \mathrm{cm}$ from the contact point of
the forearm bone and the upper-arm bone
(humerus). The triceps muscle pulls vertically upward on the forearm at distance $d_{2}=2.5 \mathrm{cm}$ behind that
contact point. Distance $d_{3}$ is 35 $\mathrm{cm} .$ What are the (a) magnitude and
(b) direction (up or down) of the force on the forearm from the tri-
ceps muscle and the (c) magnitude and (d) direction (up or down) of
the force on the forearm from the humerus?

Keshav Singh

Numerade Educator

Problem 28

In Fig. $12-45,$ suppose the length $L$ of the uniform bar is 3.00 $\mathrm{m}$
and its weight is 200 $\mathrm{N}$ . Also, let the
block's weight $W=300 \mathrm{N}$ and the an-
gle $\theta=30.0^{\circ} .$ The wire can withstand
a maximum tension of 500 $\mathrm{N.}$ (a) What is the maximum possible distance $x$
before the wire breaks? With the
block placed at this maximum $x,$ what
are the (b) horizontal and (c) vertical
components of the force on the bar
from the hinge at $A ?$

Brandy Heflin

Numerade Educator

Problem 29

A door has a height of 2.1 $\mathrm{m}$ along a $y$ axis that extends vertically
upward and a width of 0.91 $\mathrm{m}$ along an
$x$ axis that extends outward from the
hinged edge of the door. A hinge 0.30 $\mathrm{m}$ from the top and a hinge 0.30 $\mathrm{m}$ from
the bottom each support half the door's
mass, which is 27 $\mathrm{kg}$ . In unit-vector
notation, what are the forces on the
door at (a) the top hinge and (b) the
bottom hinge?

Keshav Singh

Numerade Educator

Problem 30

In Fig. $12-46,$ a 50.0 kg uniform square sign, of edge length $L=2.00 \mathrm{m},$ is
hung from a horizontal rod of length
$d_{h}=3.00 \mathrm{m}$ and negligible mass. A cable is attached to the end of the rod and to a point on the wall at distance $d_{y}=4.00 \mathrm{m}$ above the point
where the rod is hinged to the wall. (a) What is the tension in the
cable? What are the (b) magnitude and (c) direction (left or right) of the horizontal component of the force on the rod from the wall,
and the (d) magnitude and (e) direction (up or down) of the vertical component of this force?

Averell Hause

Carnegie Mellon University

Problem 31

In Fig. $12-47, \quad \mathrm{a}$ nonuniform bar is suspended
at rest in a horizontal position
by two massless cords. One
cord makes the angle $\theta=$
$36.9^{\circ}$ with the vertical; the other makes the angle $\phi=$
$53.1^{\circ}$ with the vertical. If the
length $L$ of the bar is $6.10 \mathrm{m},$
compute the distance $x$ from the left end of the bar to its center of mass.

Keshav Singh

Numerade Educator

Problem 32

In Fig. $12-48,$ the driver of a car on a horizontal road makes
an emergency stop by applying the brakes so that all four wheels
lock and skid along the road. The coefficient of kinetic friction between tires and road is $0.40 .$ The separation between the front and
rear axles is $L=4.2 \mathrm{m},$ and the center of mass of the car is located at distance $d=1.8 \mathrm{m}$ behind the front axle and distance $h=0.75 \mathrm{m}$
above the road. The car weighs 11 $\mathrm{kN}$ . Find the magnitude of $(\mathrm{a})$
the braking acceleration of the car, (b) the normal force on each rear wheel, (c) the normal force on each front wheel, (d) the braking force on each rear wheel, and (e) the braking force on each
front wheel. (Hint: Although the car is not in translational equilibrium, it is in rotational equilibrium.)

Averell Hause

Carnegie Mellon University

Problem 33

Figure $12-49 a$ shows a vertical uniform beam of length $L$ that is hinged at its lower end. A horizontal force $\vec{F}_{a}$ is applied to the beam at distance $y$ from the lower end. The beam remains
vertical because of a cable attached at the upper end, at angle $\theta$
with the horizontal. Figure $12-49 b$ gives the tension $T$ in the cable
as a function of the position of the applied force given as a fraction $y / L$ of the beam length. The scale of the $T$ axis is set by $T_{s}=600 \mathrm{N}$ .
Figure $12-49 \mathrm{cgives}$ the magnitude $F_{h}$ of the horizontal force on the beam from the hinge, also as a function of $y / L$ . Evaluate (a) angle $\theta$
and $(b)$ the magnitude of $\vec{F}_{a} .$

Keshav Singh

Numerade Educator

Problem 34

In Fig. $12-45,$ a thin horizontal bar $A B$ of negligible weight and length $L$ is hinged to a vertical wall at $A$ and supported at $B$
by a thin wire $B C$ that makes an angle $\theta$ with the horizontal. A
block of weight $W$ can be moved anywhere along the bar; its position is defined by the distance $x$ from the wall to its center of
mass. As a function of $x,$ find (a) the tension in the wire, and the
(b) horizontal and (c) vertical components of the force on the bar
from the hinge at $A .$

Averell Hause

Carnegie Mellon University

Problem 35

A cubical box is filled with sand and weighs 890 N. We wish to "roll" the box by pushing horizontally on one of the
upper edges.(a) What minimum force is required? (b) What minimum coefficient of static friction between box and floor is required? (c) If there is a more efficient way to roll the box, find the smallest possible force that would have to be applied directly to
the box to roll it. (Hint: At the onset of tipping, where is the normal
force located?

Keshav Singh

Numerade Educator

Problem 36

Figure $12-50$ shows a 70 $\mathrm{kg}$ climber hanging by only the crimp hold of one hand on
the edge of a shallow horizontal ledge in a
rock wall. (The fingers are pressed down to
gain purchase.) Her feet touch the rock wall at distance $H=2.0 \mathrm{m}$ directly below her
crimped fingers but do not provide any support. Her center of mass is distance $a=0.20$
$\mathrm{m}$ from the wall. Assume that the force from
the ledge supporting her fingers is equally shared by the four fingers. What are the values
of the (a) horizontal component $F_{h \text { and }}(b)$
vertical component $F_{v}$ of the force on each
fingertip?

Averell Hause

Carnegie Mellon University

Problem 37

In Fig. $12-51,$ a uniform plank, with a length $L$ of 6.10 $\mathrm{m}$ and a weight of 445 $\mathrm{N}$ , rests
on the ground and against a frictionless roller at
the top of a wall of height $h=3.05 \mathrm{m}$ . The plank
remains in equilibrium for any value of $\theta \geq 70^{\circ}$
but slips if $\theta<70^{\circ} .$ Find the coefficient of static
friction between the plank and the ground.

Keshav Singh

Numerade Educator

Problem 38

In Fig. $12-52,$ uniform beams $A$ and $B$ are attached to a wall with hinges
and loosely bolted together (there is
no torque of one on the other). Beam
$A$ has length $L_{A}=2.40 \mathrm{m}$ and mass
$54.0 \mathrm{kg} ;$ beam $B$ has mass 68.0 $\mathrm{kg} .$ The two hinge points are separated by distance $d=1.80 \mathrm{m} .$ In unit-vector notation, what is the force on (a) beam $A$
due to its hinge, (b) beam $A$ due to the bolt, (c) beam $B$ due to its hinge,
and $(\mathrm{d})$ beam $B$ due to the bolt?

Averell Hause

Carnegie Mellon University

Problem 39

For the stepladder shown in Fig. $12-53,$ sides $A C$ and $C E$ are each
2.44 m long and hinged at $C .$ Bar $B D$
is a tie-rod 0.762 $\mathrm{m}$ long, halfway up.
A man weighing 854 $\mathrm{N}$ climbs 1.80 $\mathrm{m}$
along the ladder. Assuming that the floor is frictionless and neglecting the
mass of the ladder, find (a) the tension
in the tie-rod and the magnitudes of
the forces on the ladder from the floor at $(\mathrm{b}) A$ and $(\mathrm{c}) E .$ (Hint: Isolate parts
of the ladder in applying the equilibrium conditions.)

Keshav Singh

Numerade Educator

Problem 40

Figure $12-54 a$ shows a horizontal uniform beam of mass $m_{b}$ and
length $L$ that is supported on the left by a hinge attached to a wall and on the right by a cable at angle $\theta$ with
the horizontal. A package of mass $m_{p}$ is positioned on the beam at a
distance $x$ from the left end. The total mass is $m_{b}+m_{p}=61.22 \mathrm{kg}$ . Figure $12-54 b$ gives the tension $T$ in the cable as a function of the
package's position given as a fraction $x / L$ of the beam length. The
scale of the $T$ axis is set by $T_{a}=500 \mathrm{N}$ and $T_{b}=700 \mathrm{N}$ . Evaluate (a)
angle $\theta,(\mathrm{b})$ mass $m_{b},$ and (c) mass $m_{p}$ .

Averell Hause

Carnegie Mellon University

Problem 41

A crate, in the form of a cube with edge lengths of $1.2 \mathrm{m},$ contains a piece of machinery; the center of mass of the crate and its
contents is located 0.30 $\mathrm{m}$ above the crate's geometrical center. The
crate rests on a ramp that makes an angle $\theta$ with the horizontal. As $\theta$
is increased from zero, an angle will be reached at which the crate will either tip over or start to slide down the ramp. If the coefficient
of static friction $\mu_{s}$ between ramp and crate is $0.60,$ (a) does the crate
tip or slide and (b) at what angle $\theta$ does this occur? If $\mu_{s}=0.70,$ (c) does the crate tip or slide and (d) at what angle $\theta$ does this occur?
(Hint: At the onset of tipping, where is the normal force located?)

Keshav Singh

Numerade Educator

Problem 42

In Fig. $12-7$ and the associated sample problem, let the coefficient of static friction $\mu_{s}$ between the ladder and the pavement be $0.53 .$ How far (in percent) up the ladder must the firefighter go to put the ladder on the verge of sliding?

Averell Hause

Carnegie Mellon University

Problem 43

A horizontal aluminum rod 4.8 $\mathrm{cm}$ in diameter projects 5.3 $\mathrm{cm}$ from a wall. A 1200 $\mathrm{kg}$ object is suspended from the
end of the rod. The shear modulus of aluminum is $3.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$ .
Neglecting the rod's mass, find (a) the shear stress on the rod and
(b) the vertical deflection of the end of the rod.

Keshav Singh

Numerade Educator

Problem 44

Figure $12-55$ shows the stress - strain curve for a material.
The scale of the stress axis is set by
$s=300$ , in units of $10^{6} \mathrm{N} / \mathrm{m}^{2} .$ What
are (a) the Young's modulus and (b)
the approximate yield strength for
this material?

Averell Hause

Carnegie Mellon University

Problem 45

In Fig. $12-56,$ a lead brick rests horizontally on cylinders $A$ and $B$ .
The areas of the top faces of the cylinders are related by $A_{A}=2 A_{B} ;$ the
Young's moduli of the cylinders
are related by $E_{A}=2 E_{B} .$ The cylinders had identical lengths before the brick was placed on them. What
fraction of the brick's mass is supported (a) by cylinder $A$ and $(b)$ by
cylinder $B ?$ The horizontal distances between the center of mass of the
brick and the centerlines of the
cylinders are $d_{A}$ for cylinder $A$ and $d_{B}$ for cylinder $B .(\mathrm{c})$ What is the ratio $d_{A} / d_{B} ?$

Keshav Singh

Numerade Educator

Problem 46

Figure $12-57$ shows an approximate plot of stress versus strain for a spider-web thread, out to the point of breaking at a
strain of $2.00 .$ The vertical axis scale is set by values $a=0.12$
GN/m $^{2}, b=0.30$ GN/m $^{2}$ , and $c=0.80$ GN/m? Assume that the thread has an initial length of $0.80 \mathrm{cm},$ an initial cross-sectional area of
$8.0 \times 10^{-12} \mathrm{m}^{2},$ and (during stretching) a constant volume. The
strain on the thread is the ratio of the change in the thread's
length to that initial length, and the stress on the thread is the ratio of the collision force to that initial cross-sectional area.
Assume that the work done on the thread by the collision force is
given by the area under the curve on the graph. Assume also that
when the single thread snares a flying insect, the insect's kinetic energy is transferred to the stretching of the thread. (a) How
much kinetic energy would put the thread on the verge of break-
ing? What is the kinetic energy of (b) a fruit fly of mass 6.00 $\mathrm{mg}$
and speed 1.70 $\mathrm{m} / \mathrm{s}$ and $(\mathrm{c})$ a bumble bee of mass 0.388 $\mathrm{g}$ and speed 0.420 $\mathrm{m} / \mathrm{s} ?$ Would (d) the fruit fly and (e) the bumble bee break the thread?

Averell Hause

Carnegie Mellon University

Problem 47

A tunnel of length $L=150 \mathrm{m},$ height $H=7.2 \mathrm{m},$ and width 5.8 $\mathrm{m}$ (with a flat roof) is to be constructed at distance $d=60 \mathrm{m}$
beneath the ground. (See Fig. $12-58 . )$ The tunnel roof is to be supported entirely by square steel columns, each with a cross-sectional
area of 960 $\mathrm{cm}^{2} .$ The mass of 1.0 $\mathrm{cm}^{3}$ of the ground material is 2.8 $\mathrm{g}$ . (a) What is the total weight of the ground material the columns must
support? (b) How many columns are needed to keep the compressive stress on each column at one-half its ultimate strength?

Surendra Kumar

Numerade Educator

Problem 48

Figure $12-59$ shows the stress versus strain plot for an
aluminum wire that is stretched
by a machine pulling in opposite
directions at the two ends of the
wire. The scale of the stress axis is set by $s=7.0,$ in units of
$10^{7} \mathrm{N} / \mathrm{m}^{2} .$ The wire has an initial
length of 0.800 $\mathrm{m}$ and an initial
cross-sectional area of $2.00 \times 10^{-6}$
$\mathrm{m}^{2} .$ How much work does the force from the machine do on the wire to produce a strain of $1.00 \times 10^{-3} ?$

Averell Hause

Carnegie Mellon University

Problem 49

In Fig. $12-60,$ a 103 kg uniform log hangs by two steel wires,
$A$ and $B,$ both of radius 1.20 $\mathrm{mm}$ .
Initially, wire $A$ was 2.50 $\mathrm{m}$ long
and 2.00 $\mathrm{mm}$ shorter than wire $B$ .
The log is now horizontal. What
are the magnitudes of the forces
on it from (a) wire $A$ and (b) wire
$B ?(\mathrm{c})$ What is the ratio $d_{A} / d_{B} ?$

Keshav Singh

Numerade Educator

Problem 50

Figure $12-61$ represents an insect caught at the midpoint of a spider-web thread. The
thread breaks under a stress of
$8.20 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ and a strain of
2.00 . Initially, it was horizontal and had a length of 2.00 $\mathrm{cm}$ and a
cross-sectional area of 8.00$x$ $10^{-12} \mathrm{m}^{2} .$ As the thread was stretched under the weight of the insect, its volume remained constant. If the weight of the insect puts the thread on the verge of breaking, what is the insect's
mass? (A spider's web is built to break if a potentially harmful insect, such as a bumble bee, becomes snared in the web.)

Averell Hause

Carnegie Mellon University

Problem 51

Figure $12-62$ is an overhead view of a rigid rod that turns about a vertical axle until the identical rubber stoppers $A$ and $B$ are forced against rigid walls at distances $r_{A}=7.0 \mathrm{cm}$ and $r_{B}=4.0 \mathrm{cm}$
from the axle. Initially the stoppers touch the walls without being
compressed. Then force $\vec{F}$ of magnitude 220 $\mathrm{N}$ is applied perpendicular to the rod at a distance $R=5.0 \mathrm{cm}$ from the axle. Find the magnitude of the force compressing (a) stopper $A$ and (b) stopper $B$ .

Keshav Singh

Numerade Educator

Problem 52

After a fall, a 95 kg rock climber finds himself dangling from the end of a rope that had been 15 $\mathrm{m}$ long and 9.6 $\mathrm{mm}$ in diameter
but has stretched by 2.8 $\mathrm{cm} .$ For the rope, calculate (a) the strain,
(b) the stress, and (c) the Young's modulus.

Averell Hause

Carnegie Mellon University

Problem 53

In Fig. $12-63,$ a rectangular slab of slate rests on a bedrock surface inclined at angle $\theta=26^{\circ} .$ The slab has length $L=43 \mathrm{m},$ thickness
$T=2.5 \mathrm{m},$ and width $W=12 \mathrm{m},$ and
1.0 $\mathrm{cm}^{3}$ of it has a mass of 3.2 $\mathrm{g} .$ The coefficient of static friction between slab and bedrock is $0.39 .$ (a) Calculate the component of the
gravitational force on the slab parallel to the bedrock surface. (b)
Calculate the magnitude of the static frictional force on the slab.
By comparing (a) and (b), you can see that the slab is in danger of
sliding. This is prevented only by chance protrusions of bedrock.
(c) To stabilize the slab, bolts are to be driven perpendicular to the
bedrock surface (two bolts are shown). If each bolt has a cross-sectional area of 6.4 $\mathrm{cm}^{2}$ and will snap under a shearing stress of
$3.6 \times 10^{8} \mathrm{N} / \mathrm{m}^{2},$ what is the minimum number of bolts needed? Assume that the bolts do not affect the normal force.

Keshav Singh

Numerade Educator

Problem 54

A uniform ladder whose length is 5.0 $\mathrm{m}$ and whose weight is 400 $\mathrm{N}$ leans against a frictionless vertical wall. The coefficient of static friction
between the level ground and the foot of the ladder is $0.46 .$ What is the
greatest distance the foot of the ladder can be placed from the base of
the wall without the ladder immediately slipping?

Averell Hause

Carnegie Mellon University

Problem 55

In Fig. $12-64,$ block $A$ (mass 10 $\mathrm{kg}$ ) is in equilibrium, but it
would slip if block $B$ (mass 5.0 $\mathrm{kg} )$ were any heavier. For angle $\theta=30^{\circ}$ what is the coefficient of static friction between block $A$ and the sur-
face below it?

Keshav Singh

Numerade Educator

Problem 56

Figure $12-65 a$ shows a uniform ramp between two buildings that allows for motion between the buildings due to strong winds. At its left end, it is hinged to the building wall; at its right end, it has
a roller that can roll along the building wall. There is no vertical
force on the roller from the building, only a horizontal force with
magnitude $F_{h}$ The horizontal distance between the buildings is
$D=4.00 \mathrm{m} .$ The rise of the ramp is $h=0.490 \mathrm{m} . \mathrm{A}$ man walks across the ramp from the left. Figure $12-65 b$ gives $F_{h}$ as a function
of the horizontal distance $x$ of the man from the building at the
left. The scale of the $F_{h}$ axis is set by $a=20 \mathrm{kN}$ and $b=25 \mathrm{kN}$ . What are the masses of (a) the ramp and (b) the man?

Averell Hause

Carnegie Mellon University

Problem 57

In Fig. $12-66,$ a 10 kg sphere is supported on a frictionless plane
inclined at angle $\theta=45^{\circ}$ from the
horizontal. Angle $\phi$ is $25^{\circ} .$ Calculate the tension in the cable.

Keshav Singh

Numerade Educator

Problem 58

In Fig. $12-67 a$ , a uniform 40.0 $\mathrm{kg}$
beam is centered over two rollers.
Vertical lines across the beam mark
off equal lengths. Two of the lines are
centered over the rollers; a 10.0 $\mathrm{kg}$
package of tamales is centered over
roller $B$ . What are the magnitudes of
the forces on the beam from (a)
roller $A$ and $(b)$ roller $B ?$ The beam
is then rolled to the left until the
right-hand end is centered over
roller $B($ Fig. $12-67 b) .$ What now are
the magnitudes of the forces on the
beam from (c) roller $A$ and $(\mathrm{d})$
roller $B ?$ Next, the beam is rolled to
the right. Assume that it has a
length of 0.800 $\mathrm{m}$ .
(e) What horizontal distance between the
package and roller $B$ puts the beam on
the verge of losing contact with
roller $A$ ?

Averell Hause

Carnegie Mellon University

Problem 59

In Fig. $12-68,$ an 817 $\mathrm{kg}$
construction bucket is suspended
by a cable $A$ that is attached at $O$
to two other cables $B$ and $C,$ making angles $\theta_{1}=51.0^{\circ}$ and $\theta_{2}=66.0^{\circ}$
with the horizontal. Find the tensions in
(a) cable $A,$ (b) cable $B$ ,
and (c) cable $C .$ (Hint: To avoid
solving two equations in two unknowns,
position the axes as
shown in the figure.)

Keshav Singh

Numerade Educator

Problem 60

In Fig. $12-69$ , a package of mass
$m$ hangs from a short cord that is tied
to the wall via cord 1 and to the ceiling
via cord $2 .$ Cord 1 is at angle $\phi=$
$40^{\circ}$ with the horizontal; cord 2 is at
angle $\theta .($ a) For what value of $\theta$ is the
tension in cord 2 minimized? (b) In
terms of $m g,$ what is the minimum tension in cord 2$?$

Averell Hause

Carnegie Mellon University

Problem 61

The force $\vec{F}$ in Fig. $12-70$
keeps the 6.40 $\mathrm{kg}$ block and the pulleys in
equilibrium. The pulleys have negligible
mass and friction. Calculate the tension $T$ in
the upper cable. (Hint: When a cable wraps
halfway around a pulley as here, the
magnitude of its net force on the pulley is twice
the tension in the cable.)

Keshav Singh

Numerade Educator

Problem 62

A mine elevator is supported by a single
steel cable 2.5 $\mathrm{cm}$ in diameter. The total
mass of the elevator cage and occupants is
670 $\mathrm{kg} .$ By how much does the cable stretch
when the elevator hangs by (a) 12 $\mathrm{m}$ of
cable and (b) 362 $\mathrm{m}$ of cable? (Neglect the
mass of the cable.)

Averell Hause

Carnegie Mellon University

Problem 63

Four bricks of length $L,$ identical and
uniform, are stacked on top of one
another (Fig. $12-71 )$ in such a way that
part of each extends beyond the
one beneath. Find, in terms of
L, the maximum values of
(a) $a_{1},(\mathrm{b}) a_{2},(\mathrm{c}) a_{3},(\mathrm{d})$
$a_{4},$ and $(\mathrm{e}) h,$ such
that the stack is in equilibrium, on the verge of falling.

Keshav Singh

Numerade Educator

Problem 64

In Fig. $12-72,$ two identical, uniform, and frictionless spheres, each of
mass $m,$ rest in a rigid rectangular container. A line connecting their centers is
at $45^{\circ}$ to the horizontal. Find the magnitudes of the forces on the spheres
from (a) the bottom of the container, (b) the left side of the container,
(c) the right side of the container, and (d) each other. (Hint: The force of one
sphere on the other is directed along the center-center line.)

Averell Hause

Carnegie Mellon University

Problem 65

In Fig. $12-73,$ a uniform beam with a weight of 60 $\mathrm{N}$ and a length of
3.2 $\mathrm{m}$ is hinged at its lower end, and a horizontal force $\vec{F}$ of magnitude
50 $\mathrm{N}$ acts at its upper end. The beam is held vertical by a cable that makes
angle $\theta=25^{\circ}$ with the ground and is attached to the beam at height $h=$
2.0 $\mathrm{m} .$ What are (a) the tension in the cable and (b) the force on the
beam from the hinge in unit-vector notation?

Keshav Singh

Numerade Educator

Problem 66

A uniform beam is 5.0 $\mathrm{m}$ long and has a mass of 53 $\mathrm{kg} .$ In Fig. $12-$ $74,$ the beam is supported in a horizontal position by a hinge and a cable, with angle $\theta=60^{\circ} .$ In unit-vector notation, what is the force on the beam from the hinge?

Averell Hause

Carnegie Mellon University

Problem 67

A solid copper cube has an edge length of 85.5 $\mathrm{cm} .$ How much stress must be applied to the cube to reduce the edge length to 85.0 $\mathrm{cm} ?$ The bulk modulus of copper is $1.4 \times 10^{11} \mathrm{N} / \mathrm{m}^{2}$.

Keshav Singh

Numerade Educator

Problem 68

A construction worker attempts to lift a uniform beam off the
floor and raise it to a vertical position. The beam is 2.50 $\mathrm{m}$ long and
weighs 500 $\mathrm{N} .$ At a certain instant the worker holds the beam momentarily
at rest with one end at distance $d=$ 1.50 $\mathrm{m}$ above the floor, as shown in
Fig. $12-75,$ by exerting a force $\vec{P}$ on the beam, perpendicular to the
beam. (a) What is the magnitude $P ?$ (b) What is the magnitude of the (net) force of the floor on the beam? (c) What is the minimum value the coefficient of static
friction between beam and floor can have in order for the beam not to slip at this instant?

Averell Hause

Carnegie Mellon University

Problem 69

In Fig. $12-76,$ a uniform rod of mass $m$ is hinged to a building at its lower end, while its upper end is held in place by a rope attached to the wall. If
angle $\theta_{1}=60^{\circ},$ what value must angle $\theta_{2}$ have so
that the tension in the rope is equal to $m g / 2 ?$

Keshav Singh

Numerade Educator

Problem 70

A 73 $\mathrm{kg}$ man stands on a level bridge of length $L .$ He is at distance $L / 4$ from one end. The bridge is uniform and weighs 2.7 $\mathrm{kN}$ . What are the
magnitudes of the vertical forces on the bridge from its supports at (a) the end farther from him and (b) the nearer end?

Averell Hause

Carnegie Mellon University

Problem 71

A uniform cube of side length 8.0 $\mathrm{cm}$ rests on a horizontal floor. The coefficient of static friction between cube and floor is $\mu .$ A horizontal pull $\vec{P}$ is applied perpendicular to one of the vertical faces of the cube, at a distance 7.0 $\mathrm{cm}$ above the floor on the vertical midline of the cube face. The magnitude of $\vec{P}$ is gradually increased. During that increase, for what values of $\mu$ will the cube eventually (a) begin to slide and (b) begin to tip? (Hint: At the onset of tipping, where is the normal force located?)

Keshav Singh

Numerade Educator

Problem 72

The system in Fig. $12-77$ is in equilibrium. The angles are $\theta_{1}=60^{\circ}$ and $\theta_{2}=20^{\circ},$ and the ball has mass $M=2.0 \mathrm{kg} .$ What is the tension
in (a) string $a b$ and (b) string $b c ?$

Vishal Gupta

Numerade Educator

Problem 73

A uniform ladder is 10 $\mathrm{m}$ long and weighs 200 $\mathrm{N} .$ In
Fig. $12-78$ , the ladder leans against a vertical, frictionless wall at height
$h=8.0 \mathrm{m}$ above the ground. $\mathrm{A}$ horizontal force $\vec{F}$ is applied to the ladder at distance $d=2.0 \mathrm{m}$ from its base (measured along the ladder). (a) If force magnitude $F=50$ $\mathrm{N},$ what is the force of the ground
on the ladder, in unit-vector notation? (b) If $F=150 \mathrm{N},$ what is the
force of the ground on the ladder, also in unit-vector notation? (c) Suppose the coefficient of static friction between the ladder and the ground is $0.38 ;$ for what minimum value of the force magnitude $F$ will the base of the ladder just barely start to move toward the wall?

Keshav Singh

Numerade Educator

Problem 74

A pan balance is made up of a rigid, massless rod with a hanging pan attached at each end. The rod is supported at and free to rotate about a point not at its center. It is balanced by unequal masses placed in the two pans. When an unknown mass $m$ is placed in the left pan, it is balanced by a mass $m_{1}$ placed in the right pan; when the mass $m$ is placed in the right pan, it is balanced by a mass $m_{2}$ in the left pan. Show that $m=\sqrt{m_{1} m_{2}}$.

Averell Hause

Carnegie Mellon University

Problem 75

The rigid square frame in Fig. $12-79$ consists of the four side bars
$A B, B C, C D,$ and $D A$ plus two diagonal bars $A C$ and $B D,$ which pass each
other freely at $E .$ By means of the turnbuckle $G,$ bar $A B$ is put under tension,
as if its ends were subject to horizontal, outward forces $\vec{T}$ of magnitude 535 $\mathrm{N}$ . (a) Which of the other bars are in tension? What are the magnitudes of (b)
the forces causing the tension in those bars and (c) the forces causing compression in the other bars? (Hint: Symmetry considerations can lead to considerable simplification in this problem.)

Keshav Singh

Numerade Educator

Problem 76

A gymnast with mass 46.0 $\mathrm{kg}$ stands on the end of a uniform balance beam as shown in Fig. $12-80 .$ The beam is 5.00 $\mathrm{m}$ long and has a mass of 250 $\mathrm{kg}$ (excluding the mass of the two supports). Each support is 0.540 $\mathrm{m}$ from its end of the beam. In unit-vector notation, what are the forces on the beam due to (a) support 1 and (b) support 2$?$

Averell Hause

Carnegie Mellon University

Problem 77

Figure $12-81$ shows a 300 $\mathrm{kg}$ cylinder that is horizontal. Three
steel wires support the cylinder from a ceiling. Wires 1 and 3 are attached at the ends of the cylinder, and wire 2 is attached at the center. The wires each have a cross-sectional area of $2.00 \times 10^{-6} \mathrm{m}^{2}$. Initially (before the cylinder was put in place) wires 1 and 3 were 2.0000 $\mathrm{m}$ long and wire 2 was 6.00 $\mathrm{mm}$ longer than that. Now (with the cylinder in place) all three wires have been stretched. What is the tension in (a) wire 1 and (b) wire 2?

Keshav Singh

Numerade Educator

Problem 78

In Fig. $12-82,$ a uniform beam of length 12.0 $\mathrm{m}$ is supported by a horizontal cable and a hinge at angle $\theta=$ $50.0^{\circ} .$ The tension in the cable is 400
N. In unit-vector notation, what are (a) the gravitational force on the beam and (b) the force on the beam from the hinge?

Averell Hause

Carnegie Mellon University

Problem 79

Four bricks of length $L,$ identical and uniform, are stacked on a table in two ways, as shown in Fig. $12-83$ (compare with Problem 63). We seek to maximize the over-hang distance $h$ in both arrangements. Find the optimum distances $a_{1}, a_{2}, b_{1},$ and $b_{2},$ and calculate $h$ for the two arrangements.

Keshav Singh

Numerade Educator

Problem 80

A cylindrical aluminum rod, with an initial length of 0.8000 $\mathrm{m}$ and radius 1000.0 $\mu \mathrm{m}$ is clamped in place at one end and then stretched by a machine pulling parallel to its length at its other end. Assuming that the rod's density (mass per unit volume) does not change, find the force magnitude that is required of the machine to decrease the radius to 999.9$\mu \mathrm{m} .$ (The yield strength is not exceeded.)

Averell Hause

Carnegie Mellon University

Problem 81

A beam of length $L$ is carried by three men, one man at one end and the other two supporting the beam between them on a crosspiece placed so that the load of the beam is equally divided among the three men. How from the beam's free end is the crosspiece placed? (Neglect the mass of the crosspiece.)

Keshav Singh

Numerade Educator

Problem 82

If the (square) beam in Fig. $12-6 a$ and the associated sample problem is of Douglas fir, what must be its thickness to keep the compressive stress on it to $\frac{1}{6}$ of its ultimate strength?

Averell Hause

Carnegie Mellon University

Problem 83

Figure $12-84$ shows a stationary arrangement of two crayon boxes and three cords. Box $A$ has a mass of 11.0 $\mathrm{kg}$ and is on a ramp at angle $\theta=30.0^{\circ} ;$ box $B$ has a mass of 7.00 $\mathrm{kg}$ and hangs on a cord. The cord connected to box $A$ is parallel to the ramp, which is frictionless. (a) What is the tension in the upper cord, and (b) what angle does that cord make with the horizontal?

Keshav Singh

Numerade Educator

Problem 84

A makeshift swing is constructed by making a loop in one end of a rope and tying the other end to a tree limb. A child is sitting in the loop with the rope hanging vertically when the child's father pulls on the child with a horizontal force and displaces the child to one side. Just before the child is released from rest, the rope makes an angle of $15^{\circ}$ with the vertical and the tension in the rope is 280 $\mathrm{N}$ . (a) How much does the child weigh? (b) What is the magnitude of the (horizontal) force of the father on the child just before the child is released? (c) If the maximum horizontal force the father can exert on the child is $93 \mathrm{N},$ what is the maximum angle with the vertical the rope can make while the father is pulling horizontally?

Averell Hause

Carnegie Mellon University

Problem 85

Figure $12-85 a$ shows details of a finger in the crimp hold of the climber in Fig. $12-50 .$ A tendon that runs from muscles in the forearm is attached to the far bone in the finger. Along the way, the tendon runs through several guiding sheaths called pulleys. The A2 pulley is attached to the first finger bone; the A4 pulley is attached to the second finger bone. To pull the finger toward the palm, the forearm muscles pull the tendon through the pulleys, much like strings on a marionette can be pulled to move parts of the marionette. Figure $12-85 b$ is a simplified diagram of the second finger bone, which has length $d .$ The tendon's pull $\vec{F}_{t}$ on the bone acts at the point where the tendon enters the A4 pulley, at distance $d / 3$ along the bone. If the force components on each of the four crimped fingers in Fig. $12-50$ are $F_{h}=13.4 \mathrm{N}$ and $F_{v}=$
$162.4 \mathrm{N},$ what is the magnitude of $\vec{F}_{t} ?$ The result is probably tolerable, but if the climber hangs by only one or two fingers, the A2 and A4 pulleys can be ruptured, a common ailment among rock climbers.

Keshav Singh

Numerade Educator

Problem 86

A trap door in a ceiling is 0.91 $\mathrm{m}$ square, has a mass of 11 $\mathrm{kg}$, and is hinged along one side, with a catch at the opposite side. If the center of gravity of the door is 10 $\mathrm{cm}$ toward the hinged side from the door's center, what are the magnitudes of the forces exerted by the door on (a) the catch and (b) the hinge?

Averell Hause

Carnegie Mellon University

Problem 87

A particle is acted on by forces given, in newtons, by $\vec{F}_{1}=$ $8.40 \hat{\mathrm{i}}-5.70 \hat{\mathrm{j}}$ and $\vec{F}_{2}=16.0 \hat{\mathrm{i}}+4.10 \hat{\mathrm{j}}$ . (a) What are the $x$ component and (b) $y$ component of the force $\vec{F}_{3}$ that balances the sum of these forces? (c) What angle does $\vec{F}_{3}$ have relative to the $+x$ axis?

Keshav Singh

Numerade Educator

Problem 88

The leaning Tower of Pisa is 59.1 $\mathrm{m}$ high and 7.44 $\mathrm{m}$ in diameter. The top of the tower is displaced 4.01 $\mathrm{m}$ from the vertical. Treat the tower as a uniform, circular cylinder. (a) What additional displacement, measured at the top, would bring the tower to the verge of toppling? (b) What angle would the tower then make with the vertical?

Averell Hause

Carnegie Mellon University