University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 11

Static Equilibrium - all with Video Answers

Educators

Chapter Questions

Problem 1

A $3.0-\mathrm{kg}$ broom is leaning against a coffee table. $A$ woman lifts the broom handle with her arm fully stretched so that her hand is a distance of $0.45 \mathrm{~m}$ from her shoulder What torque is produced on her shoulder if her arm is at an angle of $50^{\circ}$ below the horizontal?
a) $7.0 \mathrm{~N} \mathrm{~m}$
c) $8.5 \mathrm{~N} \mathrm{~m}$
b) $5.8 \mathrm{~N} \mathrm{~m}$
d) $10.1 \mathrm{~N} \mathrm{~m}$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 2

A uniform beam of mass $M$ and length $L$ is held in static equilibrium, and so the magnitude of the net torque about its center of mass is zero. The magnitude of the net torque on this beam at one of its ends, a distance of $L / 2$ from the center of mass, is
a) $M g L$.
c) zero.
b) $M g L / 2$.
d) $2 M g L$.

Narayan Hari

Numerade Educator

Problem 3

A very light and rigid rod is pivoted at point $A$ and weights $m_{1}$ and $m_{2}$ are hanging from it, as shown in the figure. The ratio of the weight of $m_{1}$ to that of $m_{2}$ is 1: 2 . What is the ratio of $L_{1}$ to $L_{2},$ the distances from the pivot point to $m_{1}$ and $m_{2}$, respectively?
a) 1: 2
b) 2: 1
c) 1: 1
d) not enough information to determine

Narayan Hari

Numerade Educator

Problem 4

Which of the following are in static equilibrium?
a) a pendulum at the top of its swing
b) a merry-go-round spinning at constant angular velocity
c) a projectile at the top of its trajectory (with zero velocity)
d) all of the above
e) none of the above

Sachin Rao

Numerade Educator

Problem 5

The object in the figure below is suspended at its center of mass-thus, it is balanced. If the object is cut in two pieces at its center of mass, what is the relation between the two resulting masses?
a) The masses are equal.
b) $M_{1}$ is less than $M_{2}$.
c) $M_{2}$ is less than $M_{1}$.
d) It is impossible to tell.

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 6

As shown in the figure, two weights are hanging on a uniform wooden bar that is $60 \mathrm{~cm}$ long and has a mass of $100 \mathrm{~g} .$ Is this system in equilibrium?
a) yes
b) no
c) cannot be
determined
d) depends on the value of the normal force

Narayan Hari

Numerade Educator

Problem 7

11.7 A $15-\mathrm{kg}$ child sits on a playground seesaw, $2.0 \mathrm{~m}$ from the pivot. A second child located $1.0 \mathrm{~m}$ on the other side of the pivot would have to have a mass of _________ to lift the first child off the ground.
a) greater than $30 \mathrm{~kg} \quad$ c) equal to $30 \mathrm{~kg}$
b) less than $30 \mathrm{~kg}$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 8

A mobile is constructed from a metal bar and two wooden blocks as shown in the figure. The metal bar has a mass of $1 \mathrm{~kg}$ and is $10 \mathrm{~cm}$ long. The metal bar has a $3-\mathrm{kg}$ wooden block hanging from the left end and a string tied to it at a distance of $3 \mathrm{~cm}$ from the left end. What mass should the wooden block hanging from the right end of the bar have to keep the bar level?
a) $0.7 \mathrm{~kg}$
b) $0.8 \mathrm{~kg}$
c) $0.9 \mathrm{~kg}$
d) $1.0 \mathrm{~kg}$
e) $1.3 \mathrm{~kg}$
f) $3.0 \mathrm{~kg}$
g) $7.0 \mathrm{~kg}$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 9

There are three sets of landing gear on an airplane: One main set is located under the centerline of each wing, and the third set is located beneath the nose of the plane. Each set of main landing gear has four tires, and the nose landing gear has two tires. If the load on all of the tires is the same when the plane is at rest, find the center of mass of the plane. Express your result as a fraction of the perpendicular distance between the centerline of the plane's wings to the plane's nose gear. (Assume that the landing gear struts are vertical when the plane is at rest and the dimensions of the landing gear are negligible compared to the dimensions of the plane.)

Keshav Singh

Numerade Educator

Problem 10

A semicircular arch of radius $a$ stands on level ground as shown in the figure. The arch is uniform in cross section and density, with total weight $W$. By symmetry, at the top of the arch, each of the two legs exerts only horizontal forces on the other; ideally, the stress at that point is uniform compression over the cross section of the arch. What are the vertical and horizontal force components which must be supplied at the base of each leg to support the arch?

Keshav Singh

Numerade Educator

Problem 11

In the absence of any symmetry or other constraints on the forces involved, how many unknown force components can be determined in a situation of static equilibrium in each of the following cases?
a) All forces and objects lie in a plane.
b) Forces and objects are in three dimensions.
c) Forces act in $n$ spatial dimensions.

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 12

You have a meter stick that balances at the $55-\mathrm{cm}$
mark. Is your meter stick homogeneous?

Keshav Singh

Numerade Educator

Problem 13

You have a meter stick that balances at the $50-\mathrm{cm}$ mark. Is it possible for your meter stick to be inhomogeneous?

Keshav Singh

Numerade Educator

Problem 14

Why does a helicopter with a single main rotor generally have a second small rotor on its tail?

Keshav Singh

Numerade Educator

Problem 15

The system shown in the figure consists of a uniform (homogeneous) rectangular board that is resting on two identical rotating cylinders. The two cylinders rotate in opposite directions at equal angular velocities. Initially, the board is placed perfectly symmetrically relative to the center point between the two cylinders. Is this an equilibrium position for the board? If yes, is it in stable or unstable equilibrium? What happens if the board is given a very slight displacement out of the initial position?

Keshav Singh

Numerade Educator

Problem 16

If the wind is blowing strongly from the east, stable equilibrium for an open umbrella is achieved if its shaft points west. Why is it relatively easy to hold the umbrella directly into the wind (in this case, easterly) but very difficult to hold it perpendicular to the wind?

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 17

A sculptor and his assistant are carrying a wedge-shaped marble slab up a flight of stairs, as shown in the flight of stairs, as shown in the figure. The density of the marble is uniform. Both
are lifting straight up as they hold the slab completely stationary for a moment. Does the
sculptor have to exert more force than the assistant to keep the slab stationary? Explain.

Keshav Singh

Numerade Educator

Problem 18

As shown in the figure, a thin rod of mass $M$ and length $L$ is suspended by two wires - one at the left end and one twothirds of the distance from the left end to the right end. a) What is the tension in each wire?
b) Determine the mass that an object hung by a string attached to the far right-hand end of the rod would have to have for the tension in the left-hand wire to be zero.

Paul A.

California State Polytechnic University, Pomona

Problem 19

A uniform disk of mass $M_{1}$ and radius $R_{1}$ has a circular hole of radius $R_{2}$ cut out as shown in the figure.
a) Find the center of mass of the resulting object.
b) How many equilibrium positions does this object have when resting on its edge? Which ones are stable, which neutral, and which unstable?

Keshav Singh

Numerade Educator

Problem 20

Consider the system shown in the figure. If a pivot point is placed at a distance $L / 2$ from the ends of the rod of length $L$ and mass $5 M,$ the system will rotate clockwise. Thus, in order for the system to not rotate, the pivot point should be placed away from the center of the rod. In which direction from the center of the rod should the pivot point be placed? How far from the center of the rod should the pivot point be placed in order for the system not to rotate? (Treat masses $M$ and $2 M$ as point masses. $)$

Keshav Singh

Numerade Educator

Problem 21

A child has a set of blocks that are all made from the same type of wood. The blocks come in three shapes: a cube of side $L$, a piece the size of two cubes, and a piece equivalent in size to three of the cubes placed end to end. The child stacks three blocks as shown in the figure: a cube on the bottom, one of the longest blocks horizontally on top of that, and the medium-sized block placed vertically on top. The centers of each block are initially on a vertical line. How far can the top block be slid along the middle block before the middle block tips?

Keshav Singh

Numerade Educator

Problem 22

Why didn't the ancient Egyptians build their pyramids upside-down? In other words, use force and center-of-mass principles to explain why it is more advantageous to construct buildings with broad bases and narrow tops than the other way around.

Keshav Singh

Numerade Educator

Problem 23

A $1000-\mathrm{N}$ crate rests
on a horizontal floor. It is being pulled up by two vertical ropes. The left rope has a tension of $400 \mathrm{~N}$. Assuming the crate does not leave the floor, what can you say about the tension in the right rope?

Paul A.

California State Polytechnic University, Pomona

Problem 24

In preparation for a demonstration on conservation of energy, a professor attaches a 5.00 -kg bowling ball to a 4.00 -m-long rope. He pulls the ball $20.0^{\circ}$ away from the vertical and holds the ball while he discusses the physics principles involved. Assuming that the force he exerts on the ball is entirely in the horizontal direction, find the tension in the rope and the force the professor is exerting on the ball.

Narayan Hari

Numerade Educator

Problem 25

A sculptor and his assistant stop for a break as they carry a marble slab of length $L=2.00 \mathrm{~m}$ and mass $75.0 \mathrm{~kg}$ up the steps, as shown in the figure. The mass of the slab is uniformly distributed along its length. As they rest, both the sculptor and his assistant are pulling directly up on each end of the slab, which is at an angle of $30.0^{\circ}$ with respect to horizontal. What are the magnitudes of the forces that the sculptor and assistant must exert on the marble slab to keep it stationary during their break?

Keshav Singh

Numerade Educator

Problem 26

During a picnic, you and two of your friends decide to have a three-way tug-of-war, with three ropes in the middle tied into a knot. Roberta pulls to the west with $420 \mathrm{~N}$ of force; Michael pulls to the south with $610 \mathrm{~N}$. In what direction and with what magnitude of force should you pull to keep the knot from moving?

Paul A.

California State Polytechnic University, Pomona

Problem 27

The figure shows a photo of a typical merry-go-round, found at many playgrounds and a diagram giving a top view. Four children are standing on the ground and pulling on the merry-goround as indicated by the force arrows. The four forces have the magnitudes $F_{1}=104.9 \mathrm{~N}$ $F_{2}=89.1 \mathrm{~N}, F_{3}=62.8 \mathrm{~N},$ and
$F_{4}=120.7 \mathrm{~N}$. All the forces act in a tangential direction. With what force, $\vec{F},$ also in a tangential direction and acting at the black point, does a fifth child have to pull in order to prevent the merry- go-round from moving? Specify the magnitude of the force and state whether the force is acting counterclockwise or clockwise

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 28

A trapdoor on a stage has a mass of $19.2 \mathrm{~kg}$ and a width of $1.50 \mathrm{~m}$ (hinge side to handle side). The door can be treated as having uniform thickness and density. A small handle on the door is $1.41 \mathrm{~m}$ away from the hinge side. $\mathrm{A}$ rope is tied to the handle and used to raise the door. At one instant, the rope is horizontal, and the trapdoor has been partly opened so that the handle is $1.13 \mathrm{~m}$ above the floor. What is the tension, $T,$ in the rope at this time?

Keshav Singh

Numerade Educator

Problem 29

A rigid rod of mass $m_{3}$ is pivoted at point $A$, and masses $m_{1}$ and $m_{2}$ are hanging from it, as shown in the figure. a) What is the normal for acting on the pivot point?
b) What is the ratio of $L_{1}$ to $L_{2},$ where these are the distances from the pivot point to $m_{1}$ and $m_{2}$, respectively? The ratio of the weights of $m_{1}, m_{2},$ and $m_{3}$ is $1: 2: 3 .$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 30

When only the front wheels of an automobile are on a platform scale, the scale balances at $8.0 \mathrm{kN} ;$ when only the rear wheels are on the scale, it balances at $6.0 \mathrm{kN}$. What is the weight of the automobile, and how far is its center of mass behind the front axle? The distance between the axles is $2.8 \mathrm{~m}$.

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 31

By considering the torques about your shoulder, estimate the force your deltoid muscles (those on top of the shoulder) must exert on the bone of your upper arm, in order to keep your arm extended straight out at shoulder level. Then, estimate the force the muscles must exert to hold a 10.0 -lb weight at arm's length. You'll need to estimate the distance from your shoulder pivot point to the point where your deltoid muscles connect to the bone of your upper arm in order to determine the necessary forces. Assume the deltoids are the only contributing muscles.

Keshav Singh

Numerade Educator

Problem 32

A uniform, equilateral triangle of side length $2.00 \mathrm{~m}$ and weight $4.00 \cdot 10^{3} \mathrm{~N}$ is placed across a gap. One point is on the north end of the gap, and the opposite side is on the south end. Find the force on each side.

Keshav Singh

Numerade Educator

Problem 33

A 600.0-N bricklayer is $1.5 \mathrm{~m}$ from one end of a uniform scaffold that is $7.0 \mathrm{~m}$ long and weighs $800.0 \mathrm{~N}$. A pile of bricks weighing $500.0 \mathrm{~N}$ is $3.0 \mathrm{~m}$ from the same end of the scaffold. If the scaffold is supported at both ends, calculate the force on each end.

Paul A.

California State Polytechnic University, Pomona

Problem 34

The uniform rod in the figure is supported by two strings. The string attached to the wall is horizontal, and the string attached to the ceiling makes an angle of $\phi$ with respect to the vertical. The rod itself is tilted from the vertical by an angle $\theta .$ If $\phi=$ $30.0^{\circ},$ what is the value of $\theta ?$

Keshav Singh

Numerade Educator

Problem 35

A construction supervisor of mass $M=92.1 \mathrm{~kg}$ is standing on a board of mass $m=27.5 \mathrm{~kg} .$ Two sawhorses at a distance $\ell=3.70 \mathrm{~m}$ apart support the board. If the man stands a distance $x_{1}=1.07 \mathrm{~m}$ away from the left-hand sawhorse as shown in the figure, what is the force that the board exerts on that sawhorse?

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 36

In a butcher shop, a horizontal steel bar of mass $4.00 \mathrm{~kg}$ and length $1.20 \mathrm{~m}$ is supported by two vertical wires attached to its ends. The butcher hangs a sausage of mass $2.40 \mathrm{~kg}$ from a hook that is at a distance of $0.20 \mathrm{~m}$ from the left end of the bar. What are the tensions in the two wires?

Keshav Singh

Numerade Educator

Problem 37

Two uniform planks, each of mass $m$ and length $L,$ are connected by a hinge at the top and by a chain of negligible mass attached at their centers, as shown in the figure. The assembly will stand upright, in the shape of an $A,$ on a frictionless surface without collapsing. As a function of the length of the chain, find each of the following:
a) the tension in the chain,
b) the force on the hinge of each plank, and
c) the force of the ground on each plank.

Keshav Singh

Numerade Educator

Problem 38

Three strings are tied together. They lie on top of a circular table, and the knot is exactly in the center of the table, as shown in the figure (top view). Each string hangs over the edge of the table, with a weight supported from it. The masses $m_{1}=4.30 \mathrm{~kg}$ and $m_{2}=$ $5.40 \mathrm{~kg}$ are known. The angle $\alpha=74^{\circ}$ between strings 1 and 2 is also known. What is the angle $\beta$ between strings 1 and $3 ?$

Keshav Singh

Numerade Educator

Problem 39

A uniform ladder $10.0 \mathrm{~m}$ long is leaning against a frictionless wall at an angle of $60.0^{\circ}$ above the horizontal. The weight of the ladder is $20.0 \mathrm{lb} . \mathrm{A} 61.0-\mathrm{lb}$ boy climbs $4.00 \mathrm{~m}$ up the ladder. What is the magnitude of the frictional force exerted on the ladder by the floor?

Keshav Singh

Numerade Educator

Problem 40

Robin is making a mobile to hang over her baby sister's crib. She purchased four stuffed animals: a teddy bear $(16.0 \mathrm{~g})$ a lamb $(18.0 \mathrm{~g})$, a little pony $(22.0 \mathrm{~g})$
and a bird $(15.0 \mathrm{~g})$. She also purchased three small wooden dowels, each $15.0 \mathrm{~cm}$ long and of mass $5.00 \mathrm{~g}$, and thread of negligible mass. She wants to hang the bear and the pony from the ends of one dowel and the lamb and the bird from the ends of the second dowel. Then, she wants to suspend the two dowels from the ends of the third dowel and hang the whole assembly from the ceiling. Explain where the thread should be attached on each dowel so that the entire assembly will hang level.

Dominador Tan

Numerade Educator

Problem 41

A door, essentially a uniform rectangle of height $2.00 \mathrm{~m}$ width $0.80 \mathrm{~m}$, and weight $100.0 \mathrm{~N}$, is supported at one edge by two hinges, one $30.0 \mathrm{~cm}$ above the bottom of the door and one $170.0 \mathrm{~cm}$ above the bottom of the door. Calculate the horizontal components of the forces on the two hinges .

Paul A.

California State Polytechnic University, Pomona

Problem 42

The figure shows a $20.0-\mathrm{kg}$, uniform ladder of length
$L$ hinged to a horizontal platform at point $P_{1}$ and anchored with a steel cable of the same length as the ladder attached at the ladder's midpoint. Calculate the tension in the cable and the forces in the hinge when an $80.0-\mathrm{kg}$ person is standing three-quarters of the way up the ladder.

Keshav Singh

Numerade Educator

Problem 43

A beam with a length of $8.00 \mathrm{~m}$ and a mass of $100 . \mathrm{kg}$ is attached by a large bolt to a support at a distance of $3.00 \mathrm{~m}$ from one end. The beam makes an angle $\theta=30.0^{\circ}$ with the horizontal, as shown in the figure. A mas $M=500 . \mathrm{kg}$ is attached with a rope to one end of the beam, and a second rope is attached at a right angle to the other end of the beam. Find the tension, $T$, in the second rope and the force exerted on the beam by the bolt.

Anand Jangid

Numerade Educator

Problem 44

A ball of mass $15.49 \mathrm{~kg}$ rests on a table of height $0.72 \mathrm{~m}$. The tabletop is a rectangular glass plate of mass $12.13 \mathrm{~kg},$ which is supported at the corners by thin legs, as shown in the figure. The width of the tabletop is $w=138.0 \mathrm{~cm},$ and its depth $d=63.8 \mathrm{~cm} .$ If the ball touches the tabletop at a point $(x, y)=(69.0 \mathrm{~cm}, 16.6 \mathrm{~cm})$ relative to corner 1 , what is the force that the tabletop exerts on each leg?

Paul A.

California State Polytechnic University, Pomona

Problem 45

A wooden bridge crossing a canyon consists of a plank with length density $\lambda=2.00 \mathrm{~kg} / \mathrm{m}$suspended at $h=10.0 \mathrm{~m}$ below atree branch by two ropes of length $L=2 h$ and with a maximum rated tension of $2000 . \mathrm{N},$ which are attached to the ends of the plank, as shown in the figure. A hiker steps onto the bridge from the left side, causing the bridge to tip to an angle of $25.0^{\circ}$ with respect to the horizontal. What is the mass of the hiker?

Aishwarya Krishnakumar

Aishwarya Krishnakumar

Numerade Educator

Problem 46

The famous Gateway Arch in St. Louis, Missouri, is approximately an inverted catenary. A simple example of such a curve is given by $y(x)=2 a-a \cosh (x / a),$ where $y$ is vertical height and $x$ is horizontal distance, measured from directly under the top of the curve; thus, $x$ varies from $-a \cosh ^{-1} 2$ to $+a \cosh ^{-1} 2$, with $a$ the height of the top of the curve (see the figure). Suppose an arch of uniform cross section and density, with total weight $W$, has this shape. The two legs of the arch exert only horizontal forces on each other at the top; ideally, the stress there should be uniform compression across the cross section.
a) Calculate the vertical and horizontal force components that are acting at the base of each leg of this arch.
b) At what angle should the bottom face of the legs be oriented?

Keshav Singh

Numerade Educator

Problem 47

A uniform rectangular bookcase of height $H$ and width $W=H / 2$ is to be pushed at a constant velocity across a level floor. The bookcase is pushed horizontally at its top edge, at the distance $H$ above the floor. What is the maximum value the coefficient of kinetic friction between the bookcase and the floor can have if the bookcase is not to tip over while being pushed?

Narayan Hari

Numerade Educator

Problem 48

The system shown in the figure is in static equilibrium. The rod of length $L$ and mass $M$ is held in an upright position. The top of the rod is tied to a fixed vertical surface by a string, and a force $F$ is applied at the midpoint of the rod. The coefficient of static friction between the rod and the horizontal surface is $\mu_{\mathrm{s}}$. What is the maximum force, $F$, that can be applied and have the rod remain in static equilibrium?

Narayan Hari

Numerade Educator

Problem 49

A ladder of mass $37.7 \mathrm{~kg}$ and length $3.07 \mathrm{~m}$ is leaning against a wall at an angle $\theta$. The coefficient of static friction between ladder and floor is 0.313 ; assume that the friction force between ladder and wall is zero. What is the maximum value that $\theta$ can have before the ladder starts slipping?

Keshav Singh

Numerade Educator

Problem 50

A uniform rigid pole of length $L$ and mass $M$ is to be supported from a vertical wall in a horizontal position, as shown in the figure. The pole is not attached directly to the wall, so the coefficient of static friction, $\mu_{\mathrm{s}}$, between the wall and the pole provides the only vertical force on one end of the .11 .50 A uniform rigid pole of length $L$ and mass $M$ is to be supported from a vertical wall in a horizontal position, as shown in the figure. The pole is not attached directly to the wall, so the coefficient of static friction, $\mu_{s}$, between the wall and the pole provides the only vertical force on one end of the the wall. Determine the minimum valueof $\mu_{s}$, as a function of $L$ and $D$, that will keep the pole horizontal and not allow its end to slide down
the wall.

Paul A.

California State Polytechnic University, Pomona

Problem 51

A boy weighing $60.0 \mathrm{lb}$ is playing on a plank. The plank weighs $30.0 \mathrm{lb}$, is uniform, is $8.00 \mathrm{ft}$ long, and lies on two supports, one $2.00 \mathrm{ft}$ from the left end and the other $2.00 \mathrm{ft}$ from the right end.
a) If the boy is $3.00 \mathrm{ft}$ from the left end, what force is exerted by each support?
b) The boy moves toward the right end. How far can he go before the plank will tip?

Keshav Singh

Numerade Educator

Problem 52

A track has a height that is a function of horizontal position $x$, given by $h(x)=x^{3}+3 x^{2}-24 x+16$. Find all the positions on the track where a marble will remain where it is placed. What kind of equilibrium exists at each of these positions?

Keshav Singh

Numerade Educator

Problem 53

The figure shows a stack of seven identical aluminum blocks each of length $l=15.9 \mathrm{~cm}$ and thickness $d=2.2 \mathrm{~cm},$ stacked on a table.
a) How far to the right of the edge of the table is it possible for the right edge of the top (seventh) block to extend?
b) What is the minimum height of a stack of these blocks for which the left edge of the top block is to the right of the right edge of the table?

Paul A.

California State Polytechnic University, Pomona

Problem 54

You are using a 5.00 -m-long ladder to paint the exterior of your house. The point of contact between the ladder and the siding of the house is $4.00 \mathrm{~m}$ above the ground. The ladder exerted by the side wall and the ground on the ladder and
(b) the coefficient of static friction between the ground and the base of the ladder that is necessary to keep the ladder stable.

Paul A.

California State Polytechnic University, Pomona

Problem 55

A ladder of mass $M$ and length $L=4.00 \mathrm{~m}$ is on a level floor leaning against a vertical wall. The coefficient of static friction between the ladder and the floor is $\mu_{\mathrm{s}}=$ 0.600 , while the friction between the ladder and the wall is negligible. The ladder is at an angle of $\theta=50.0^{\circ}$ above the horizontal. A man of mass $3 M$ starts to climb the ladder. To what distance up the ladder can the man climb before the ladder starts to slip on the floor?

Keshav Singh

Numerade Educator

Problem 56

A machinist makes the object shown in the figure. The larger-diameter cylinder is made of brass (density of $8.60 \mathrm{~g} / \mathrm{cm}^{3}$ ); the small er-diameter cylinder is made of aluminum (density of $\left.2.70 \mathrm{~g} / \mathrm{cm}^{3}\right) .$ The dimensions
are $r_{1}=2.00 \mathrm{~cm}, r_{2}=4.00$
$\mathrm{cm}, d_{1}=20.0 \mathrm{~cm},$ and $d_{2}=$
$4.00 \mathrm{~cm} .$
a) Find the location of the center of mass.
b) If the object is on its side, as shown in the figure, is it in equilibrium? If yes, is this stable equilibrium?

Paul A.

California State Polytechnic University, Pomona

Problem 57

An object is restricted to movement in one dimension. Its position is specified along the $x$ -axis. The potential energy of the object as a function of its position is given by $U(x)=a\left(x^{4}-2 b^{2} x^{2}\right),$ where $a$ and $b$ represent positive numbers. Determine the location(s) of any equilibrium point(s), and classify the equilibrium at each point as stable, unstable, or neutral.

Keshav Singh

Numerade Educator

Problem 58

A two-dimensional object with uniform mass density is in the shape of a thin square and has a mass of $2.00 \mathrm{~kg} .$ The sides of the square are each $20.0 \mathrm{~cm}$ long. The coordinate system has its origin at the center of the square. A point mass, $m,$ of $2.00 \cdot 10^{2} \mathrm{~g}$ is placed at one corner of the square object, and the assembly is held in equilibrium by positioning the support at position $(x, y),$ as shown in the figure. Find the location of the support.

Keshav Singh

Numerade Educator

Problem 59

Persons $A$ and $B$ are standing on a board of uniform linear density that is balanced on two supports, as shown in the figure. What is the maximum distance $x$ from the right end of the board at which person $A$ can stand without tipping the board? Treat persons $A$ and $B$ as point masses. The mass of person $B$ is twice that of person $A$, and the mass of the board is half that of person $A$. Give your answer in terms of $L$, the length of the board.

Keshav Singh

Numerade Educator

Problem 60

An SUV has a height $h$ and a wheelbase of length b. Its center of mass is midway between the wheels and at a distance $\alpha h$ above the ground, where $0<\alpha<1$. The SUV enters a turn at a dangerously high speed, $v$. The radius of the turn is $R(R \gg b)$, and the road is flat. The coefficient of static
friction between the road and
the properly inflated tires is $\mu_{s}$. After entering the turn, the SUV will either skid out of the
turn or begin to tip.
a) The SUV will skid out of the turn if the friction force reaches its maximum value, $F \rightarrow \mu_{\mathrm{s}} N$. Determine the speed, $v_{\text {skid }},$ for which this will occur. Assume no tipping occurs.
b) The torque keeping the SUV from tipping acts on the outside wheel. The highest value this force can have is equal to the entire normal force. Determine the speed, $v_{\text {tip }}$, at which this will occur. Assume no skidding occurs.
c) It is safer if the SUV skids out before it tips. This will occur as long as $v_{\text {skid }}<v_{\text {tip }}$. Apply this condition, and determine the maximum value for $\alpha$ in terms of $b, h$ and $\mu_{\mathrm{s}}$

Dominador Tan

Numerade Educator

Problem 61

A wooden plank with length $L=8.00 \mathrm{~m}$ and mass $M=100 . \mathrm{kg}$ is centered on a granite cube with side $S=2.00 \mathrm{~m}$ A person of mass $m=65.0 \mathrm{~kg}$ begins walking from the center of the plank outward, as shown in the figure. How far from the center of the plank does the person get before the plank starts tipping?

Keshav Singh

Numerade Educator

Problem 62

A board, with a weight $m g=120.0 \mathrm{~N}$and a length of $5.00 \mathrm{~m}$, is supported by two vertical
ropes, as shown in the figure. Rope $A$ is connected to one end of the board, and rope $B$ is connected at a distance $d=1.00 \mathrm{~m}$ from the other end of the board. A box with a weight $M g=20.0 \mathrm{~N}$ is placed on the board with its center of mass at $d=1.00 \mathrm{~m}$ from rope $A .$ What are the tensions in the two ropes?

Paul A.

California State Polytechnic University, Pomona

Problem 63

In a car, which is accelerating at $5.00 \mathrm{~m} / \mathrm{s}^{2},$ an air freshener is hanging from the rear-view mirror, with the string maintaining a constant angle with respect to the vertical. What is this angle?

Narayan Hari

Numerade Educator

Problem 64

Typical weight sets used for bodybuilding consist of disk-shaped weights with holes in the center that can slide onto $2.2-\mathrm{m}$ -long barbells. A barbell is supported by racks located a fifth of its length from each end, as shown in the figure. What is the minimum mass $m$ of the barbell if a bodybuilder is to slide a weight with $M=22 \mathrm{~kg}$ onto the end without the barbell tipping off the rack? Assume that the barbell is a uniform rod.

Paul A.

California State Polytechnic University, Pomona

Problem 65

A $5.00-\mathrm{m}$ -long board of mass $50.0 \mathrm{~kg}$ is used as a seesaw. On the left end of the seesaw sits a 45.0 -kg girl, and on the right end sits a 60.0 -kg boy. Determine the position of the pivot point for static equilibrium.

Keshav Singh

Numerade Educator

Problem 66

A mobile consists of two very lightweight rods of length $l=0.400 \mathrm{~m}$ connected to each other and the ceiling by vertical strings. (Neglect the masses of the rods and strings. Three objects are suspended by strings from the rods. The masses of objects 1 and 3 are $m_{1}=6.40 \mathrm{~kg}$ and
$m_{3}=3.20 \mathrm{~kg} .$ The
distance $x$ shown in the figure is $0.160 \mathrm{~m}$. What is the mass of $m_{2} ?$

Keshav Singh

Numerade Educator

Problem 67

In the experimental setup shown in the figure, a beam, $B_{1},$ of unknown mass $M_{1}$ and length $L_{1}=1.00 \mathrm{~m}$ is pivoted about its lowest point at $P_{1} .$ A second beam, $B_{2},$ of mass $M_{2}=0.200 \mathrm{~kg}$ and length $L_{2}=0.200 \mathrm{~m}$ is suspended (pivoted) from $B_{1}$ at a point $P_{2}$, which is a horizontal distance $d=0.550 \mathrm{~m}$ from $P_{1}$. To keep the system at equilibrium, a mass $m=0.500 \mathrm{~kg}$ has to be suspended from a massless string that runs horizontally from $P_{3}$, at the top of beam $B_{1}$, and passes over a frictionless pulley. The string runs at a vertical distance $y=0.707 \mathrm{~m}$ above the pivot point $P_{1}$. Calculate the mass of beam $B_{1}$.

Keshav Singh

Numerade Educator

Problem 68

An important characteristic of the condition of static equilibrium is the fact that the net torque has to be zero irrespective of the choice of pivot point. For the setup in Problem $11.67,$ prove that the torque is indeed zero with respect to a pivot point at $P_{1}, P_{2},$ or $P_{3}$.

Paul A.

California State Polytechnic University, Pomona

Problem 69

One end of a heavy beam of mass $M=50.0 \mathrm{~kg}$ is hinged to a vertical wall, and the other end is tied to a steel cable of length $3.0 \mathrm{~m}$, as shown in the figure. The other end of the cable is also attached to the wall at a distance of $4.0 \mathrm{~m}$ above the hinge. mass $m=20.0 \mathrm{~kg}$ is hung from one end of the beam by a rope.
a) Determine the tensions in the cable and the rope.
b) Find the force the hinge exerts on the beam.

Paul A.

California State Polytechnic University, Pomona

Problem 70

A $100 .-\mathrm{kg}$ uniform bar of length $L=5.00 \mathrm{~m}$ is attached to a wall by a hinge at point $A$ and supported in a horizontal position by a light cable attached to its other end. The cable is attached to the wall at point $B$, at a distance $D=2.00 \mathrm{~m}$ above point $A$. Find: a) the tension, $T,$ on the cable and
b) the horizontal and vertical components of the force acting on the bar at point $A$.

Keshav Singh

Numerade Educator

Problem 71

A mobile over a baby's crib displays small colorful shapes. What values for $m_{1}, m_{2}$, and $m_{3}$ are needed to keep the mobile balanced (with all rods horizontal)?

Paul A.

California State Polytechnic University, Pomona

Problem 72

Consider the rod of length $L$ shown in the figure. The mass of the rod is $m=2.00 \mathrm{~kg}$ and the pivot point is located at the left end (at $x=0$ ). In order to prevent the rod from rotating, a variable force given by $F(x)=(15.0 \mathrm{~N})(x / L)^{4}$ is applied to the rod. At what point $x$ on the rod should the force be applied in order to keep it from rotating?

Keshav Singh

Numerade Educator

Problem 73

A pipe that is $2.20 \mathrm{~m}$ long and has a mass of $8.13 \mathrm{~kg}$ is suspended horizontally over a stage by two chains, each located $0.20 \mathrm{~m}$ from an end. Two $7.89-\mathrm{kg}$ theater lights are clamped onto the pipe, one $0.65 \mathrm{~m}$ from the left end and the other $1.14 \mathrm{~m}$ from the left end. Calculate the tension in each chain.

Keshav Singh

Numerade Educator

Problem 74

A 2.00 -m-long diving board of mass $12.0 \mathrm{~kg}$ is $3.00 \mathrm{~m}$ above the water. It has two attachments holding it in place. One is located at the very back end of the board, and the other is $25.0 \mathrm{~cm}$ away from that end.
a) Assuming that the board has uniform density, find the forces acting on each attachment (take the downward direction to be positive).
b) If a diver of mass $65.0 \mathrm{~kg}$ is standing on the front end, what are the forces acting on the two attachments?

Paul A.

California State Polytechnic University, Pomona

Problem 75

A $20.0-\mathrm{kg}$ box with a height of $80.0 \mathrm{~cm}$ and a width of $30.0 \mathrm{~cm}$ has a handle on the side that is $50.0 \mathrm{~cm}$ above the ground. The box is at rest, and the coefficient of static friction between the box and the floor is 0.28
a) What is the minimum force, $F$, that can be applied to the handle so that the box will tip over without slipping?
b) In what direction should this force be applied?

Dominador Tan

Numerade Educator

Problem 76

The angular displacement of a torsional spring is proportional to the applied torque; that is $\tau=\kappa \theta,$ where $\kappa$ is a constant. Suppose that such a spring is mounted to an arm that moves in a vertical plane. The mass of the arm is $45.0 \mathrm{~g}$, and it is $12.0 \mathrm{~cm}$ long. The arm-spring system is at equilibrium with the arm at an angular displacement of $17.0^{\circ}$ with respect to the horizontal. If a mass of $0.420 \mathrm{~kg}$ is hung from the arm $9.00 \mathrm{~cm}$ from the axle, what will be the angular displacement in the new equilibrium position (relative to that with the unloaded spring)?

Aishwarya Krishnakumar

Aishwarya Krishnakumar

Numerade Educator