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Schaum’s Outline of College Physics

Eugene Hecht

Chapter 5

Equilibrium of a Rigid Body Under Coplanar Forces - all with Video Answers

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Chapter Questions

04:23

Problem 1

Imagine a bar of steel $80 \mathrm{~cm}$ long pivoted horizontally at its left end, as depicted in Fig. $5-2 .$ Find the torque about axis- $A$ (which is perpendicular to the page) due to each of the forces shown acting at its right end.

Paul Gabriel
Paul Gabriel
Numerade Educator
03:42

Problem 2

A uniform metal beam of length $L$ weighs $200 \mathrm{~N}$ and holds a $450-\mathrm{N}$ object as shown in Fig. 5-3. Find the magnitudes of the forces exerted on the beam by the two supports at its ends. Assume the lengths are exact.
Rather than draw a separate free-body diagram, we show the forces on the object being considered (the beam) in $\underline{\text { Fig. } 5-3 \text { . }}$ Because the beam is uniform, its center of gravity is at its geometric center. Thus, the weight of the beam ( $200 \mathrm{~N}$ ) is shown acting downward at the beam's center. The forces $F_{1}$ and $F_{2}$ are exerted on the beam by the supports. Because there are no $x$ directed forces acting on the beam, we have only two equations to write for this equilibrium situation: $\Sigma F_{y}=0$ and $\Sigma \tau=0$.

Paul Gabriel
Paul Gabriel
Numerade Educator
01:52

Problem 3

A uniform, horizontal, 100-N pipe is used as a lever, as shown in Fig. $5-4 .$ Where must the fulcrum (the support point) be placed if a 500-N weight at one end is to balance a 200-N weight at the other end? What is the upward reaction force exerted by the support on the pipe?
The forces in question are shown in $\underline{\text { Fig. } 5-4}$, where $F_{R}$ is the reaction force of the support on the pipe. The weight of the pipe acts downward at its center. We assume that the support point is at a distance $x$ from one end. Take the axis of rotation to be at the support point. Then the torque equation, $\Omega+\sum \tau=0$, about that point becomes

Paul Gabriel
Paul Gabriel
Numerade Educator
03:03

Problem 4

Where must a $0.80-\mathrm{kN}$ object be hung on a uniform, horizontal, rigid 100-N pole so that a girl pushing up at one end supports onethird as much as a woman pushing up at the other end?
The situation is shown in Fig. $5-5$, where the weight of the pole acts down at its center. We represent the force exerted by the girl as $F$, and that by the woman as $3 F$. There are two unknowns, $F$ and $x$, and we will need two equations. To avoid the possibility of writing equations that turn out not to be independent, it's a good practice to write one sum-of-the-torques equation and one sum-ofthe-forces equation. Take the rotational axis point at the left end. Then the torque equation becomes

Paul Gabriel
Paul Gabriel
Numerade Educator
02:35

Problem 5

A uniform, horizontal, $0.20-\mathrm{kN}$ board of length $L$ has two objects hanging from it with weights of $300 \mathrm{~N}$ at exactly $L / 3$ from one end and $400 \mathrm{~N}$ at exactly $3 L / 4$ from the same end. What single additional force acting on the board will cause the board to be in equilibrium?

The situation is drawn in $\underline{\text { Fig. } 5-6, \text { where } F \text { is the force we wish to }}$ find. For equilibrium, $\Sigma F_{y}=0$ and so
$$
F=400 \mathrm{~N}+200 \mathrm{~N}+300 \mathrm{~N}=900 \mathrm{~N}
$$Because the board is to be in equilibrium, we are free to locate the axis of rotation anywhere. Choose it at point- $A$ at the left end of the board, since all the forces are measured (as to location) from that end in the diagram. Then $\Sigma \tau=0$, and taking counterclockwise as positive,
$+(x)(F)\left(\sin 90^{\circ}\right)-(3 L / 4)(400 \mathrm{~N})\left(\sin 90^{\circ}\right)-(L / 2)(200 \mathrm{~N})\left(\sin 90^{\circ}\right)-(L / 3)$
$(300 \mathrm{~N})\left(\sin 90^{\circ}\right)=0$
Using $F=900 \mathrm{~N}$, we find that $x=0.56 L$. The required force is $0.90 \mathrm{kN}$ upward at $0.56 L$ from the left end.

Paul Gabriel
Paul Gabriel
Numerade Educator
03:34

Problem 6

The right-angle rule (or square) depicted in Fig. $5-7$ hangs at rest from a peg as shown. It is made of a uniform metal sheet. One arm is $L \mathrm{~cm}$ long, while the other is $2 L \mathrm{~cm}$ long. Find (to two significant figures) the angle $\theta$ at which it will hang.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:51

Problem 7

Consider the situation illustrated in Fig. $5-8(a)$. The uniform $0.60$ $\mathrm{kN}$ beam is hinged at $P$. Find the tension in the tie rope and the components of the reaction force exerted by the hinge on the beam. Give your answers to two significant figures.
The reaction forces acting on the beam are shown in $\underline{\text { Fig. }} 5-8(b)$, where the force exerted by the hinge is represented by its horizontal and vertical components, $F_{R H}$ and $F_{R V}$. The torque equation about $P$ is
$\Omega[$
(We take the axis at $P$ because then $F_{R H}$ and $F_{R V}$ do not appear in the torque equation.) Solving this equation yields $F_{T}=2280 \mathrm{~N}$ or,

Paul Gabriel
Paul Gabriel
Numerade Educator
07:01

Problem 8

A uniform, $0.40-\mathrm{kN}$ boom is supported as shown in $\underline{\text { Fig. }} 5-9(a)$. Find the tension in the tie rope and the force exerted on the boom by the pin at $P$.

The forces acting on the boom are shown in $\underline{\text { Fig. }} 5-9(b)$. Take the pin as the axis of rotation. The torque equation is then

Paul Gabriel
Paul Gabriel
Numerade Educator
03:11

Problem 9

As indicated in $\underline{\text { Fig. } 5-10}$, hinges $A$ and $B$ hold a uniform, $400-\mathrm{N}$ door in place. If the upper hinge happens to support the entire weight of the door, find the forces exerted on the door at both hinges. The width of the door is exactly $h / 2$, where $h$ is the distance between the hinges.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:04

Problem 10

A ladder leans against a smooth wall, as can be seen in Fig. 5-11. (By a "smooth" wall, we mean that the wall exerts on the ladder only a force that is perpendicular to the wall. There is no friction force.) The ladder weighs $200 \mathrm{~N}$, and its center of gravity is $0.40 L$ from the base, where $L$ is the ladder's length. (a) How large a friction force must exist at the base of the ladder if it is not to slip?
(b) What is the necessary coefficient of static friction?
(a) We wish to find the friction force $F_{f}$. Notice that no friction force exists at the top of the ladder. Taking torques about point- $A$ gives the torque equation

Paul Gabriel
Paul Gabriel
Numerade Educator
12:57

Problem 11

For the situation drawn in $\underline{\text { Fig. }} 5-12(a)$, find $F_{T 1}, F_{T 2}$, and $F_{T 3}$. The boom is uniform and weighs $800 \mathrm{~N}$.

First apply the force condition to point- $A$. The appropriate freebody diagram is shown in Fig. $5-12(b)$. We then have
$F_{T 2} \cos 50.0^{\circ}-2000 \mathrm{~N}=0 \quad$ and $\quad F_{T_{1}}-F_{T 2} \sin 50.0^{\circ}=0$
From the first of these we find $F_{T 2}=3.11 \mathrm{kN}$; then the second equation gives $F_{T 1}=2.38 \mathrm{kN}$
Let us now isolate the boom and apply the equilibrium conditions to it. The appropriate free-body diagram is found in Fig. $5-12(c)$. The torque equation, for torques taken about point- $C$, is
$\bigcap\left[\sum \tau_{c}=+(L)\left(F_{n}\right)\left(\sin 20,0^{\circ}\right)-(L)(3110 \mathrm{~N})\left(\sin 90.0^{\circ}\right)-\ldots / 2 \times 800 \mathrm{~N}\right.$
Solving for $F_{T 3}$, we compute it to be $9.84 \mathrm{kN}$. If it were required, we could find $F_{R H}$ and $F_{R V}$ by using the $x$ - and $y$ -force equations.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
01:02

Problem 12

A steering wheel has a diameter of $40.0 \mathrm{~cm} .$ A force of $30.0 \mathrm{~N}$ is applied to its rim on the right, tangent to the wheel and in the plane of it. Determine the size of the resulting torque. [Hint: The moment arm is the radius. Watch out for units.]

Paul Gabriel
Paul Gabriel
Numerade Educator
01:02

Problem 13

A wrench is $50.0 \mathrm{~cm}$ long. It is placed on a nut, and a force of 100 $\mathrm{N}$ is applied perpendicular to the wrench handle. This force is in the plane of the wrench and nut, at a distance of $30.0 \mathrm{~cm}$ from the center of the nut. Determine the size of the torque twisting the nut. [Hint: Draw a diagram and label the moment arm. Watch out for units.]

Paul Gabriel
Paul Gabriel
Numerade Educator
01:15

Problem 14

A horizontal essentially weightless lever is pivoted so it can rotate freely in a vertical plane. A downward force of $30.0 \mathrm{~N}$ is applied perpendicularly to the lever at a point $25.0 \mathrm{~cm}$ from and to the right of the pivot. Determine the torque on the lever, about the pivot. [Hint: Draw a diagram and specify the direction of the torque.]

Paul Gabriel
Paul Gabriel
Numerade Educator
02:06

Problem 15

A horizontal essentially weightless lever is pivoted at its center so it can rotate freely in a vertical plane. A downward force of $80.0 \mathrm{~N}$ is applied perpendicularly to the lever at a point $35.0 \mathrm{~cm}$ from and to the right of the pivot. Another downward force of $100.0 \mathrm{~N}$ is applied perpendicularly to the lever at a point $15.0 \mathrm{~cm}$ from and to the left of the pivot. Determine the net torque on the lever. [Hint:
Draw a diagram.]

Paul Gabriel
Paul Gabriel
Numerade Educator
02:38

Problem 16

A seesaw is $5.00 \mathrm{~m}$ long with a fulcrum at its center. The uniform plank is balanced horizontally when a $40.0$ -kg kid sits at the very end on the right and an $80.0$ -kg kid sits somewhere on the left. Locate that second kid. [Hint: Draw a diagram.]

Paul Gabriel
Paul Gabriel
Numerade Educator
02:05

Problem 17

A force of $1000 \mathrm{~N}$ is applied downward at the right end of a $1.50-\mathrm{m}$ long, essentially weightless horizontal crowbar. The bar is pivoted on a rock $1.25 \mathrm{~m}$ from the right end. What is the maximum
amount of weight that can be supported on the left end before the bar moves? [Hint: Draw a diagram. Watch out for significant figures.]

Paul Gabriel
Paul Gabriel
Numerade Educator
02:20

Problem 18

An essentially weightless shovel is $120 \mathrm{~cm}$ long. Someone holds it horizontally, supporting it with his left hand at the shovel's center of gravity and his right hand $80.0 \mathrm{~cm}$ to the right of the $\mathrm{c.g}$. The shovel contains a 20.0-N rock whose c.g. is $8.00 \mathrm{~cm}$ to the right of the edge of the shovel. How much force does the person exert down on the handle? [Hint: Draw a diagram and take the torques around the left hand to avoid the force of the left hand.]

Paul Gabriel
Paul Gabriel
Numerade Educator
02:01

Problem 19

An 800 -N painter stands on a uniform horizontal 100-N plank resting on the rungs of two separated stepladders. The plank is $4.00 \mathrm{~m}$ long, and it is supported at its very ends (not a very safe arrangement). The painter stands on the plank $1.00 \mathrm{~m}$ from its right end. Determine the upward force exerted by the ladder on the left. [Hint: Draw a diagram and locate the weight of the plank at its c. $g$. and take the torques around the right end.]

Massimo Antonelli
Massimo Antonelli
Numerade Educator
06:10

Problem 20

As depicted in Fig. $5-13$, two people sit in a car that weighs 8000
N. The person in front weighs $700 \mathrm{~N}$, while the one in the back weighs $900 \mathrm{~N}$. Call $L$ the distance between the front and back wheels. The car's center of gravity is a distance $0.400 L$ behind the front wheels. How much force does each front wheel and each back wheel support if the people are seated along the centerline of the car?

Morgan Cheatham
Morgan Cheatham
Numerade Educator
01:09

Problem 21

Two people, one at each end of a uniform beam that weighs $400 \mathrm{~N}$, hold the beam at an angle of $25.0^{\circ}$ to the horizontal. How large a vertical force must each person exert on the beam?

Paul Gabriel
Paul Gabriel
Numerade Educator
06:12

Problem 22

Repeat Problem $5.13$ if a 140-N child sits on the beam at a point one-fourth of the way along the beam from its lower end.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
07:36

Problem 23

Shown in Fig. 5-14 is a uniform 1600-N beam hinged at one end and held by a horizontal tie rope at the other. Determine the tension $F_{T}$ in the rope and the force components at the hinge.

Paul Gabriel
Paul Gabriel
Numerade Educator
08:31

Problem 24

The uniform horizontal beam illustrated in Fig. $5-15$ weighs $500 \mathrm{~N}$ and supports a 700-N load. Find the tension in the tie rope and the reaction force of the hinge on the beam.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:21

Problem 25

The arm drawn in Fig. 5-16 supports a 4.0-kg sphere. The mass of the hand and forearm together is $3.0 \mathrm{~kg}$ and its weight acts at a point $15 \mathrm{~cm}$ from the elbow. Assuming all the forces are vertical, determine the force exerted by the biceps muscle.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:21

Problem 26

The mobile depicted in Fig. 5-17 hangs in equilibrium. It consists of objects held by vertical strings. Object-3 weighs $1.40 \mathrm{~N}$, while each of the identical uniform horizontal bars weighs $0.50 \mathrm{~N}$. Find
(a) the weights of objects-1 and $-2$, and $(b)$ the tension in the upper string.

Paul Gabriel
Paul Gabriel
Numerade Educator
03:36

Problem 27

The hinges of a uniform door which weighs $200 \mathrm{~N}$ are $2.5 \mathrm{~m}$ apart. One hinge is a distance $d$ from the top of the door, while the other is a distance $d$ from the bottom. The door is $1.0 \mathrm{~m}$ wide. The weight of the door is supported by the lower hinge. Determine the forces exerted by the hinges on the door.

Paul Gabriel
Paul Gabriel
Numerade Educator
07:54

Problem 28

The uniform bar in Fig. $5-18$ weighs $40 \mathrm{~N}$ and is subjected to the forces shown. Find the magnitude, location, and direction of the force needed to keep the bar in equilibrium.

Vishal Gupta
Vishal Gupta
Numerade Educator
07:12

Problem 29

The horizontal, uniform, $120-\mathrm{N}$ board drawn in Fig. $5-19$ is supported by two ropes as shown. A $0.40-\mathrm{kN}$ weight is suspended one-quarter of the way from the left end. Find $F_{T 1}, F_{T 2}$, and the

Paul Gabriel
Paul Gabriel
Numerade Educator
03:48

Problem 30

The foot of a ladder rests against a wall, and its top is held by a horizontal tie rope, as indicated in $\underline{\text { Fig. } 5-20 . \text { The ladder weighs }}$ $100 \mathrm{~N}$, and its center of gravity is $0.40$ of its length from the foot. A 150-N child hangs from a rung that is $0.20$ of the length from the top. Determine the tension in the tie rope and the components of the force on the foot of the ladder.

Paul Gabriel
Paul Gabriel
Numerade Educator
07:20

Problem 31

A truss is made by hinging two uniform, 150 -N rafters as depicted in Fig. 5-21. They rest on an essentially frictionless floor and are held together by a horizontal tie rope. A 500-N load is held at their apex. Find the tension in the tie rope.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
03:45

Problem 32

A 900-N lawn roller is to be pulled over a 5.0-cm high curb (see Fig. 5-22). The radius of the roller is $25 \mathrm{~cm}$. What minimum pulling force is needed if the angle $\theta$ made by the handle is $(a) 0^{\circ}$ and (b) $30^{\circ}$ ? [Hint: Find the force needed to keep the roller balanced against the edge of the curb, just clear of the ground.]

Chitra Gondi
Chitra Gondi
Numerade Educator
01:39

Problem 33

In Fig. 5-23, the uniform horizontal beam weighs $500 \mathrm{~N}$. If the tie rope can support $1800 \mathrm{~N}$, what is the maximum value the $\operatorname{load} F_{W}$ can have?

Paul Gabriel
Paul Gabriel
Numerade Educator
08:49

Problem 34

The beam in Fig. 5-24 has negligible weight. If the system hangs in equilibrium when $F_{W 1}=500 \mathrm{~N}$, what is the value of $F_{W 2}$ ?

Morgan Cheatham
Morgan Cheatham
Numerade Educator
03:09

Problem 35

Repeat $\underline{\text { Problem } 5.26}$, but now find $F_{W 1}$ if $F_{W 2}$ is $500 \mathrm{~N}$. Here the beam weighs $300 \mathrm{~N}$ and is uniform.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
10:57

Problem 36

An object is subjected to the forces shown in Fig. $5-25 .$ What single force $F$ applied at a point on the $x$ -axis will balance these forces leaving the object motionless? (First find its components, and then find the force.) Where on the $x$ -axis should the force be

Morgan Cheatham
Morgan Cheatham
Numerade Educator
02:43

Problem 37

The solid uniform disk of radius $b$ illustrated in Fig. $5-26$ can turn freely on an axle through its center. A hole of diameter $D$ is drilled through the disk; its center is a distance $r$ from the axle. The weight of the material drilled out is $F_{W h} .(a)$ Find the weight $F_{W}$ of an object hung from a string wound on the disk that will hold the disk in equilibrium in the position shown. (b) What would happen if the load $F_{W}$ vanished? Explain your answer.

Lindsay Williams
Lindsay Williams
Numerade Educator