Vector Mechanics for Engineers: Statics and Dynamics

Ferdinand P. Beer, E. Russell Johnston, Jr., David F. Mazurek

Chapter 3

Rigid Bodies: Equivalent Systems of Forces - all with Video Answers

Educators

Chapter Questions

Problem 1

A foot valve for a pneumatic system is hinged at $B$. Knowing that $\alpha=$ $28^{\circ},$ determine the moment of the $16-\mathrm{N}$ force about point $B$ by resolving the force into horizontal and vertical components.

Prashant Bana

Prashant Bana

Numerade Educator

Problem 2

A foot valve for a pneumatic system is hinged at $B$. Knowing that $\alpha=$ $28^{\circ},$ determine the moment of the $16-\mathrm{N}$ force about point $B$ by resolving the force into components along $A B C$ and in a direction perpendicular to $A B C$.

Prashant Bana

Numerade Educator

Problem 3

It is known that a vertical force of $200 \mathrm{lb}$ is required to remove the nail at $C$ from the board. As the nail first starts moving, determine $(a)$ the moment about $B$ of the force exerted on the nail, $(b)$ the magnitude of the force $\mathbf{P}$ that creates the same moment about $B$ if $\alpha=10^{\circ},(c)$ the smallest force $\mathbf{P}$ that creates the same moment about $B$.

AS

Anderson Speed

Numerade Educator

Problem 4

A $300-\mathrm{N}$ force is applied at $A$ as shown. Determine $(a)$ the moment of the $300-\mathrm{N}$ force about $D,(b)$ the smallest force applied at $B$ that creates the same moment about $D$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 5

A $300-\mathrm{N}$ force is applied at $A$ as shown. Determine $(a)$ the moment of the $300-\mathrm{N}$ force about $D,(b)$ the magnitude and sense of the horizontal force applied at $C$ that creates the same moment about $D,(c)$ the smallest force applied at $C$ that creates the same moment about $D .$

Imranul Haque

Numerade Educator

Problem 6

An 8 -lb force $\mathbf{P}$ is applied to a shift lever. Determine the moment of $\mathbf{P}$ about $B$ when $\alpha$ is equal to $25^{\circ}$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 7

For the shift lever shown, determine the magnitude and the direction of the smallest force $\mathbf{P}$ that has a $210-\mathrm{lb}$ -in. clockwise moment about $B$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 8

An 11 -lb force $P$ is applied to a shift lever. The moment of $P$ about $B$ is clockwise and has a magnitude of $250 \mathrm{lb} \cdot \mathrm{in} .$ Determine the value of $\alpha$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 9

$\operatorname{Rod} A B$ is held in place by the cord $A C$. Knowing that the tension in the cord is $1350 \mathrm{~N}$ and that $c=360 \mathrm{~mm}$, determine the moment about $B$ of the force exerted by the cord at point $A$ by resolving that force into horizontal and vertical components applied $(a)$ at point $A,(b)$ at point $C$

Prashant Bana

Numerade Educator

Problem 10

$\operatorname{Rod} A B$ is held in place by the cord $A C$. Knowing that $c=840 \mathrm{~mm}$ and that the moment about $B$ of the force exerted by the cord at point $A$ is $756 \mathrm{~N} \cdot \mathrm{m},$ determine the tension in the cord.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 11

The tailgate of a car is supported by the hydraulic lift $B C$. If the lift exerts a 125 -lb force directed along its centerline on the ball and socket at $B$, determine the moment of the force about $A$.

Kenny Mesadieu

Numerade Educator

Problem 12

The tailga$128.2 \mathrm{lb} \cdot \mathrm{ft} \circlearrowleft$te of a car is supported by the hydraulic lift $B C$. If the lift exerts a 125 -lb force directed along its centerline on the ball and socket at $B$, determine the moment of the force about $A$.

Kenny Mesadieu

Numerade Educator

Problem 13

It is known that the connecting $\operatorname{rod} A B$ exerts on the crank $B C$ a $2.5-\mathrm{kN}$ force directed down and to the left along the centerline of $A B$. Determine the moment of the force about $C$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 14

It is known that the connecting $\operatorname{rod} A B$ exerts on the crank $B C$ a $2.5-\mathrm{kN}$ force directed down and to the left along the centerline of $A B$. Determine the moment of the force about $C$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 15

Form the vector product $\mathbf{P}_{1} \times \mathbf{P}_{2}$ and use the result obtained to prove the identity
$$
\sin \left(\theta_{1}-\theta_{2}\right)=\sin \theta_{1} \cos \theta_{2}-\cos \theta_{1} \sin \theta_{2}
$$

Tony Ni

Numerade Educator

Problem 16

The vectors $\mathbf{P}$ and $\mathbf{Q}$ are two adjacent sides of a parallelogram. Determine the area of the parallelogram when $(a) \mathbf{P}=-2 \mathbf{i}+2 \mathbf{j}-6 \mathbf{k}$ and $\mathbf{Q}=8 \mathbf{i}+2 \mathbf{j}+4 \mathbf{k},(b) \mathbf{P}=3 \mathbf{i}-9 \mathbf{j}-7 \mathbf{k}$ and $\mathbf{Q}=-4 \mathbf{i}+2 \mathbf{j}-5 \mathbf{k}$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 17

A plane contains the vectors $\mathbf{A}$ and $\mathbf{B}$. Determine the unit vector normal to the plane when $\mathbf{A}$ and $\mathbf{B}$ are equal to, respectively, $(a) 7 \mathbf{i}+8 \mathbf{j}$ $-2 \mathbf{k}$ and $9 \mathbf{i}-4 \mathbf{j}-5 \mathbf{k},(b) 6 \mathbf{i}-3 \mathbf{j}+9 \mathbf{k}$ and $-5 \mathbf{i}+4 \mathbf{j}-3 \mathbf{k}$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 18

A line passes through the points $(-4 \mathrm{~m},-3 \mathrm{~m})$ and $(2 \mathrm{~m}, 7 \mathrm{~m})$. Determine the perpendicular distance $d$ from the line to the origin $O$ of the system of coordinates.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 19

Determine the moment about the origin $O$ of the force $\mathbf{F}=-2 \mathbf{i}-3 \mathbf{j}+$ 5k that acts at a point $A$. Assume that the position vector of $A$ is $(a) \mathbf{r}=\mathbf{i}+\mathbf{j}+\mathbf{k},(b) \mathbf{r}=4 \mathbf{i}+6 \mathbf{j}-10 \mathbf{k}$ $(c) \mathbf{r}=4 \mathbf{i}+3 \mathbf{j}-5 \mathbf{k}$

AM

Aditya Mahendru

Numerade Educator

Problem 20

Determine the moment about the origin $O$ of the force $\mathbf{F}=4 \mathbf{i}+10 \mathbf{j}+$ 6k that acts at a point $A$. Assume that the position vector of $A$ is
$(a) \mathbf{r}=2 \mathbf{i}-3 \mathbf{j}+4 \mathbf{k},(b) \mathbf{r}=2 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k},(c) \mathbf{r}=2 \mathbf{i}+5 \mathbf{j}+6 \mathbf{k}$

AM

Aditya Mahendru

Numerade Educator

Problem 21

Before the trunk of a large tree is felled, cables $A B$ and $B C$ are attached as shown. Knowing that the tensions in cables $A B$ and $B C$ are $555 \mathrm{~N}$ and $660 \mathrm{~N}$, respectively, determine the moment about $O$ of the resultant force exerted on the tree by the cables at $B$.

JM

Joseph Martinez

Numerade Educator

Problem 22

The 12 -ft boom $A B$ has a fixed end $A$. A steel cable is stretched from the free end $B$ of the boom to a point $C$ located on the vertical wall. If the tension in the cable is $380 \mathrm{lb},$ determine the moment about $A$ of the force exerted by the cable at $B$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

A 200-N force is applied as shown to the bracket $A B C$. Determine the moment of the force about $A$.

Ahmed Kamel

Numerade Educator

Problem 24

A force $\mathbf{P}$ of magnitude $200 \mathrm{~N}$ acts along the diagonal $B C$ of the bent plate shown. Determine the moment of $\mathbf{P}$ about point $E$.

Vipender Yadav

Numerade Educator

Problem 25

A 6-ft-long fishing rod $A B$ is securely anchored in the sand of a beach. After a fish takes the bait, the resulting force in the line is $6 \mathrm{lb}$. Determine the moment about $A$ of the force exerted by the line at $B$.

Fatima Iqbal

Numerade Educator

Problem 26

A precast concrete wall section is temporarily held by two cables as shown. Knowing that the tension in cable $B D$ is $900 \mathrm{~N},$ determine the moment about point $O$ of the force exerted by the cable at $B$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 27

In Prob. $3.22,$ determine the perpendicular distance from point $A$ to cable $B C$.

Kaitlin Coad

Numerade Educator

Problem 28

In Prob. 3.23 , determine the perpendicular distance from point $A$ to the line of action of the $200-\mathrm{N}$ force.

Surendra Kumar

Numerade Educator

Problem 29

In Prob. $3.24,$ determine the perpendicular distance from point $E$ to the line of action of force $\mathbf{P}$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 30

In Prob. $3.25,$ determine the perpendicular distance from point $A$ to a line drawn through points $B$ and $C$.

Rashmi Sinha

Numerade Educator

Problem 31

In Prob. 3.25 , determine the perpendicular distance from point $D$ to a line drawn through points $B$ and $C$.

Rashmi Sinha

Numerade Educator

Problem 32

In Prob. 3.26 , determine the perpendicular distance from point $O$ to cable $B D$.

BM

Brandon Matherne

Numerade Educator

Problem 33

In Prob. 3.26 , determine the perpendicular distance from point $C$ to cable $B D$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 34

Determine the value of $a$ that minimizes the perpendicular distance from point $C$ to a section of pipeline that passes through points $A$ and $B$.

Angela Guo

Numerade Educator

Problem 35

Given the vectors $\mathbf{P}=2 \mathbf{i}+\mathbf{j}+2 \mathbf{k}, \mathbf{Q}=3 \mathbf{i}+4 \mathbf{j}-5 \mathbf{k},$ and $\mathbf{S}=-4 \mathbf{i}+$
$\mathbf{j}-2 \mathbf{k},$ compute the scalar products $\mathbf{P} \cdot \mathbf{Q}, \mathbf{P} \cdot \mathbf{S},$ and $\mathbf{Q} \cdot \mathbf{S}$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 36

Form the scalar product $\mathbf{B} \cdot \mathbf{C}$ and use the result obtained to prove the identity
$$
\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta
$$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 37

Three cables are attached to the top of the tower at $A$. Determine the angle formed by cables $A B$ and $A C$.

Jordan Gassaway

Jordan Gassaway

Numerade Educator

Problem 38

Three cables are attached to the top of the tower at $A .$ Determine the angle formed by cables $A D$ and $A B$.

Jordan Gassaway

Jordan Gassaway

Numerade Educator

Problem 39

Knowing that the tension in cable $A C$ is 280 lb, determine $(a)$ the angle between cable $A C$ and the boom $A B,(b)$ the projection on $A B$ of the force exerted by cable $A C$ at point $A$.

Pawan Yadav

Numerade Educator

Problem 40

Knowing that the tension in cable $A D$ is $180 \mathrm{lb},$ determine $(a)$ the angle between cable $A D$ and the $\mathrm{boom} A B,(b)$ the projection on $A B$ of the force exerted by cable $A D$ at point $A$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 41

Ropes $A B$ and $B C$ are two of the ropes used to support a tent. The two ropes are attached to a stake at $B$. If the tension in rope $A B$ is $540 \mathrm{~N}$, determine $(a)$ the angle between rope $A B$ and the stake, $(b)$ the projection on the stake of the force exerted by rope $A B$ at point $B$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 42

Ropes $A B$ and $B C$ are two of the ropes used to support a tent. The two ropes are attached to a stake at $B$. If the tension in rope $B C$ is $490 \mathrm{~N}$, determine $(a)$ the angle between rope $B C$ and the stake, $(b)$ the projection on the stake of the force exerted by rope $B C$ at point $B$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 43

The 20 -in. tube $A B$ can slide along a horizontal rod. The ends $A$ and $B$ of the tube are connected by elastic cords to the fixed point $C$. For the position corresponding to $x=11$ in., determine the angle formed by the two cords, ( $a$ ) using Eq. $(3.30),(b)$ applying the law of cosines to triangle $A B C$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 44

Solve Prob. 3.43 for the position corresponding to $x=4$ in.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 45

Determine the volume of the parallelepiped of Fig. $3.20 b$ when
(a) $\mathbf{P}=2 \mathbf{i}-4 \mathbf{j}+3 \mathbf{k}, \mathbf{Q}=-5 \mathbf{i}-\mathbf{j}+2 \mathbf{k},$ and $\mathbf{S}=\mathbf{i}-7 \mathbf{j}-\mathbf{k}$
(b) $\mathbf{P}=6 \mathbf{i}-5 \mathbf{j}-2 \mathbf{k}, \mathbf{Q}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k},$ and $\mathbf{S}=-4 \mathbf{i}-3 \mathbf{j}+2 \mathbf{k}$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 46

Given the vectors $\mathbf{P}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{Q}=5 \mathbf{i}+\mathbf{j}-2 \mathbf{k},$ and $\mathbf{S}=\mathbf{i}+3 \mathbf{j}+S_{z} \mathbf{k}$
determine the value of $S_{z}$ for which the three vectors are coplanar.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 47

A crane is oriented so that the end of the 25 -m boom $A O$ lies in the $y z$ plane. At the instant shown, the tension in cable $A B$ is $4 \mathrm{kN}$. Determine the moment about each of the coordinate axes of the force exerted on $A$ by cable $A B$.

Sheh Lit Chang

University of Washington

Problem 48

The $25-\mathrm{m}$ crane boom $A O$ lies in the $y z$ plane. Determine the maximum permissible tension in cable $A B$ if the absolute value of moments about the coordinate axes of the force exerted on $A$ by cable $A B$ must be
$$
\left|M_{x}\right| \leq 60 \mathrm{kN} \cdot \mathrm{m},\left|M_{y}\right| \leq 12 \mathrm{kN} \cdot \mathrm{m},\left|M_{z}\right| \leq 8 \mathrm{kN} \cdot \mathrm{m}
$$

Vipender Yadav

Numerade Educator

Problem 49

To loosen a frozen valve, a force $\mathbf{F}$ with a magnitude of $70 \mathrm{lb}$ is applied to the handle of the valve. Knowing that $\theta=25^{\circ}, M_{x}=-61 \mathrm{lb} \cdot \mathrm{ft},$ and $M_{z}=-43 \mathrm{lb} \cdot \mathrm{ft},$ determine $\phi$ and $d$

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 50

When a force $\mathbf{F}$ is applied to the handle of the valve shown, its moments about the $x$ and $z$ axes are $M_{x}=-77 \mathrm{lb} \cdot \mathrm{ft}$ and $M_{z}=-81 \mathrm{lb} \cdot \mathrm{ft},$ respectively. For $d=27$ in., determine the moment $M_{y}$ of $\mathbf{F}$ about the $y$ axis.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 51

The $0.61 \times 1.00-\mathrm{m}$ lid $A B C D$ of a storage bin is hinged along side $A B$ and is held open by looping cord $D E C$ over a frictionless hook at $E$. If the tension in the cord is $66 \mathrm{~N}$, determine the moment about each of the coordinate axes of the force exerted by the cord at $D$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 52

The $0.61 \times 1.00-\mathrm{m}$ lid $A B C D$ of a storage bin is hinged along side $A B$ and is held open by looping cord $D E C$ over a frictionless hook at $E$. If the tension in the cord is $66 \mathrm{~N}$, determine the moment about each of the coordinate axes of the force exerted by the cord at $C$.

AG

Allison Gleason

Numerade Educator

Problem 53

A farmer uses cables and winch pullers $B$ and $E$ to plumb one side of a small barn. If it is known that the sum of the moments about
the $x$ axis of the forces exerted by the cables on the barn at points $A$ and $D$ is equal to $4728 \mathrm{lb} \cdot \mathrm{ft},$ determine the magnitude of $\mathbf{T}_{D E}$ when $T_{A B}=255 \mathrm{lb}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 54

Solve Prob. 3.53 when the tension in cable $A B$ is $306 \mathrm{lb}$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 55

A force $\mathbf{P}$ of magnitude $520 \mathrm{lb}$ acts on the frame shown at point $E$. Determine the moment of $\mathbf{P}$ about a line joining points $O$ and $D$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 56

A force $P$ acts on the frame shown at point $E .$ Knowing that the absolute value of the moment of $\mathbf{P}$ about a line joining points $F$ and $B$ is $300 \mathrm{lb} \cdot \mathrm{ft},$ determine the magnitude of the force $\mathbf{P}$.

Donald Albin

Numerade Educator

Problem 57

The frame $A C D$ is hinged at $A$ and $D$ and is supported by a cable that passes through a ring at $B$ and is attached to hooks at $G$ and $H$. Knowing that the tension in the cable is $450 \mathrm{~N}$, determine the moment about the diagonal $A D$ of the force exerted on the frame by portion $B H$ of the cable.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 58

In Prob. $3.57,$ determine the moment about the diagonal $A D$ of the force exerted on the frame by portion $B G$ of the cable.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 59

The triangular plate $A B C$ is supported by ball-and-socket joints at $B$ and $D$ and is held in the position shown by cables $A E$ and $C F .$ If the force exerted by cable $A E$ at $A$ is $55 \mathrm{~N},$ determine the moment of that force about the line joining points $D$ and $B$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 60

The triangular plate $A B C$ is supported by ball-and-socket joints at $B$ and $D$ and is held in the position shown by cables $A E$ and $C F .$ If the force exerted by cable $C F$ at $C$ is $33 \mathrm{~N},$ determine the moment of that force about the line joining points $D$ and $B$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 61

A regular tetrahedron has six edges of length $a$. A force $\mathbf{P}$ is directed as shown along edge $B C$. Determine the moment of $\mathbf{P}$ about edge $O A$

Sheh Lit Chang

University of Washington

Problem 62

A regular tetrahedron has six edges of length $a$. ( $a$ ) Show that two opposite edges, such as $O A$ and $B C,$ are perpendicular to each other.
(b) Use this property and the result obtained in Prob. 3.61 to determine the perpendicular distance between edges $O A$ and $B C$.

Shoukat Ali

Other Schools

Problem 63

Two forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ in space have the same magnitude $F$. Prove that the moment of $\mathbf{F}_{1}$ about the line of action of $\mathbf{F}_{2}$ is equal to the moment of $\mathbf{F}_{2}$ about the line of action of $\mathbf{F}_{1}$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 64

In Prob. 3.55 , determine the perpendicular distance between a line joining points $O$ and $D$ and the line of action of $\mathbf{P}$.

Rashmi Sinha

Numerade Educator

Problem 65

In Prob. $3.56,$ determine the perpendicular distance between a line joining points $F$ and $B$ and the line of action of $\mathbf{P}$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 66

In Prob. $3.57,$ determine the perpendicular distance between portion $B H$ of the cable and the diagonal $A D$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 67

In Prob. 3.58 , determine the perpendicular distance between portion $B G$ of the cable and the diagonal $A D$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 68

In Prob. 3.59 , determine the perpendicular distance between cable $A E$ and the line joining points $D$ and $B$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 69

In Prob. 3.60 , determine the perpendicular distance between cable $C F$ and the line joining points $D$ and $B$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 70

Two $80-\mathrm{N}$ forces are applied as shown to the corners $B$ and $D$ of a rectangular plate. ( $a$ ) Determine the moment of the couple formed by the two forces by resolving each force into horizontal and vertical components and adding the moments of the two resulting couples.
(b) Use the result obtained to determine the perpendicular distance between lines $B E$ and $D F$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 71

Two parallel $60-\mathrm{N}$ forces are applied as shown to the corners $A$ and $C$ of a 200 -mm square plate. Determine the moment of the couple formed by the two forces $(a)$ by multiplying their magnitude by their perpendicular distance, $(b)$ by resolving each force into horizontal and vertical components and adding the moments of the two resulting couples.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 72

A multiple-drilling machine is used to drill simultaneously six holes in the steel plate shown. Each drill exerts a clockwise couple of magnitude $40 \mathrm{lb} \cdot \mathrm{in} .$ on the plate. Determine an equivalent couple formed by the smallest possible forces acting $(a)$ at $A$ and $C,(b)$ at $A$ and $D,$ ( $c$ ) on the plate.

Sanu Kumar

Numerade Educator

Problem 73

Four pegs of the same diameter are attached to a board as shown. Two strings are passed around the pegs and pulled with the forces indicated. Determine the diameter of the pegs knowing that the resultant couple applied to the board is $1132.5 \mathrm{lb}$ -in. counterclockwise.

Sanu Kumar

Numerade Educator

Problem 74

A piece of plywood in which several holes are being drilled successively has been secured to a workbench by means of two nails. Knowing that the drill exerts a 12-N.m couple on the piece of plywood, determine the magnitude of the resulting forces applied to the nails if they are located $(a)$ at $A$ and $B,(b)$ at $B$ and $C,(c)$ at $A$ and $C$.

Sanu Kumar

Numerade Educator

Problem 75

The shafts of an angle drive are acted upon by the two couples shown. Replace the two couples with a single equivalent couple, specifying its magnitude and the direction of its axis.

Sanu Kumar

Numerade Educator

Problem 76

If $P=0$ in the figure, replace the two remaining couples with a single equivalent couple, specifying its magnitude and the direction of its axis.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 77

If $P=20 \mathrm{lb}$ in the figure, replace the three couples with a single equivalent couple, specifying its magnitude and the direction of its axis.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 78

The two couples shown are to be replaced with a single equivalent couple. Determine $(a)$ the couple vector representing the equivalent couple, (b) the two forces acting at $B$ and $C$ that can be used to form that couple.

Sanu Kumar

Numerade Educator

Problem 79

Solve part $a$ of Prob. 3.78 , assuming that two 15-N vertical forces have been added, one acting upward at $A$ and the other downward at $C$.

Sanu Kumar

Numerade Educator

Problem 80

Shafts $A$ and $B$ connect the gear box to the wheel assemblies of a tractor, and shaft $C$ connects it to the engine. Shafts $A$ and $B$ lie in the vertical $y z$ plane, while shaft $C$ is directed along the $x$ axis. Replace the couples applied to the shafts by a single equivalent couple, specifying its magnitude and the direction of its axis.

Victor Salazar

Numerade Educator

Problem 81

A $500-\mathrm{N}$ force is applied to a bent plate as shown. Determine $(a)$ an equivalent force-couple system at $B,(b)$ an equivalent system formed by a vertical force at $A$ and a force at $B$.

Narayan Hari

Numerade Educator

Problem 82

A crane column supports a 16 -kip load as shown. Replace the load with an equivalent system consisting of an axial force along $A B$ and a couple.

Sanu Kumar

Numerade Educator

Problem 83

A dirigible is tethered by a cable attached to its cabin at $B$. If the tension in the cable is $1040 \mathrm{~N}$, replace the force exerted by the cable at $B$ with an equivalent system formed by two parallel forces applied at $A$ and $C$.

Supratim Pal

Numerade Educator

Problem 84

A $30-\mathrm{lb}$ vertical force $\mathbf{P}$ is applied at $A$ to the bracket shown, which is held by screws at $B$ and $C$. $(a)$ Replace $\mathbf{P}$ with an equivalent forcecouple system at $B$. (b) Find the two horizontal forces at $B$ and $C$ that are equivalent to the couple obtained in part $a$.

Sanu Kumar

Numerade Educator

Problem 85

A worker tries to move a rock by applying a $360-\mathrm{N}$ force to a steel bar as shown. (a) Replace that force with an equivalent force-couple system at $D .(b)$ Two workers attempt to move the same rock by applying a vertical force at $A$ and another force at $D .$ Determine the two forces if they are to be equivalent to the single force of part $a$.

Sanu Kumar

Numerade Educator

Problem 86

A worker tries to move a rock by applying a $360-\mathrm{N}$ force to a steel bar as shown. If two workers attempt to move the same rock by applying a force at $A$ and a parallel force at $C,$ determine the two forces so that they will be equivalent to the single $360-\mathrm{N}$ force shown in the figure.

Sanu Kumar

Numerade Educator

Problem 87

The shearing forces exerted on the cross section of a steel channel can be represented by a 900-N vertical force and two $250-\mathrm{N}$ horizontal forces as shown. Replace this force and couple with a single force $\mathbf{F}$ applied at point $C,$ and determine the distance $x$ from $C$ to line $B D$. (Point $C$ is defined as the shear center of the section.)

Sanu Kumar

Numerade Educator

Problem 88

Knowing that $\alpha=60^{\circ},$ replace the force and couple shown with a single force applied at a point located $(a)$ on line $A B,(b)$ on line $C D$. In each case, determine the distance from the center $O$ to the point of application of the force.

Sanu Kumar

Numerade Educator

Problem 89

Three control rods attached to a lever $A B C$ exert on it the forces shown. $(a)$ Replace the three forces with an equivalent force-couple system at $B$. ( $b$ ) Determine the single force that is equivalent to the force-couple system obtained in part $a$, and specify its point of application on the lever.

Sanu Kumar

Numerade Educator

Problem 90

A rectangular plate is acted upon by the force and couple shown. This system is to be replaced with a single equivalent force. $(a)$ For $\alpha=40^{\circ},$ specify the magnitude and line of action of the equivalent force. (b) Specify the value of $\alpha$ if the line of action of the equivalent force is to intersect line $C D 300 \mathrm{~mm}$ to the right of $D$.

Sanu Kumar

Numerade Educator

Problem 91

While tapping a hole, a machinist applies the horizontal forces shown to the handle of the tap wrench. Show that these forces are equivalent to a single force, and specify, if possible, the point of application of the single force on the handle.

Prabhu Ramji

Numerade Educator

Problem 92

A hexagonal plate is acted upon by the force $\mathbf{P}$ and the couple shown. Determine the magnitude and the direction of the smallest force $\mathbf{P}$ for which this system can be replaced with a single force at $E$.

Sanu Kumar

Numerade Educator

Problem 93

Replace the $250-\mathrm{kN}$ force $\mathbf{P}$ with an equivalent force-couple system at $G$.

Sanu Kumar

Numerade Educator

Problem 94

A 2.6 -kip force is applied at point $D$ of the cast-iron post shown. Replace that force with an equivalent force-couple system at the center $A$ of the base section.

Sanu Kumar

Numerade Educator

Problem 95

Replace the $150-\mathrm{N}$ force with an equivalent force-couple system at $A$.

Sanu Kumar

Numerade Educator

Problem 96

To keep a door closed, a wooden stick is wedged between the floor and the doorknob. The stick exerts at $B$ a $175-\mathrm{N}$ force directed along line $A B$. Replace that force with an equivalent force-couple system at $C$.

Sanu Kumar

Numerade Educator

Problem 97

A 46-lb force $\mathbf{F}$ and a $2120-1 b$ -in. couple $\mathbf{M}$ are applied to corner $A$ of the block shown. Replace the given force-couple system with an equivalent force-couple system at corner $H$.

Sanu Kumar

Numerade Educator

Problem 98

A $110-\mathrm{N}$ force acting in a vertical plane parallel to the $y z$ plane is applied to the 220 -mm-long horizontal handle $A B$ of a socket wrench. Replace the force with an equivalent force-couple system at the origin $O$ of the coordinate system.

Sanu Kumar

Numerade Educator

Problem 99

The 12 -ft boom $A B$ has a fixed end $A,$ and the tension in cable $B C$ is 570 lb. Replace the force that the cable exerts at $B$ with an equivalent force-couple system at $A$.

Sanu Kumar

Numerade Educator

Problem 100

The jib crane shown is oriented so that its boom $A D$ is parallel to the $x$ axis and is used to move a heavy crate. Knowing that the tension in cable $A B$ is 2.6 kips, replace the force exerted by the cable at $A$ with an equivalent force-couple system at the center $O$ of the base of the crane.

M Hassan Anwar

Numerade Educator

Problem 101

A 4 -m-long beam is subjected to a variety of loadings. ( $a$ ) Replace each loading with an equivalent force-couple system at end $A$ of the beam. $(b)$ Which of the loadings are equivalent?

Sanu Kumar

Numerade Educator

Problem 102

A 4 -m-long beam is loaded as shown. Determine the loading of Prob. 3.101 that is equivalent to this loading.

Sanu Kumar

Numerade Educator

Problem 103

Determine the single equivalent force and the distance from point $A$ to its line of action for the beam and loading of $(a)$ Prob. $3.101 b,(b)$ Prob. $3.101 d,(c)$ Prob. $3.101 e$

Sanu Kumar

Numerade Educator

Problem 104

Five separate force-couple systems act at the corners of a piece of sheet metal that has been bent into the shape shown. Determine which of these systems is equivalent to a force $\mathbf{F}=(10 \mathrm{lb}) \mathbf{i}$ and a couple of moment $\mathbf{M}=(15 \mathrm{lb} \cdot \mathrm{ft}) \mathbf{j}+(15 \mathrm{lb} \cdot \mathrm{ft}) \mathrm{k}$ located at the origin.

Sanu Kumar

Numerade Educator

Problem 105

The weights of two children sitting at ends $A$ and $B$ of a seesaw are $84 \mathrm{lb}$ and $64 \mathrm{lb},$ respectively. Where should a third child sit so that the resultant of the weights of the three children will pass through $C$ if she weighs ( $a$ ) $60 \mathrm{lb},(b) 52 \mathrm{lb} ?$

Sanu Kumar

Numerade Educator

Problem 106

Three stage lights are mounted on a pipe as shown. The lights at $A$ and $B$ each weigh $4.1 \mathrm{lb},$ while the one at $C$ weighs $3.5 \mathrm{lb} .(a)$ If $d=25$ in. determine the distance from $D$ to the line of action of the resultant of the weights of the three lights. ( $b$ ) Determine the value of $d$ so that the resultant of the weights passes through the midpoint of the pipe.

Sanu Kumar

Numerade Educator

Problem 107

A beam supports three loads of given magnitude and a fourth load whose magnitude is a function of position. If $b=1.5 \mathrm{~m}$ and the loads are to be replaced with a single equivalent force, determine $(a)$ the value of $a$ so that the distance from support $A$ to the line of action of the equivalent force is maximum, $(b)$ the magnitude of the equivalent force and its point of application on the beam.

Sanu Kumar

Numerade Educator

Problem 108

A $6 \times 12$ -in. plate is subjected to four loads as shown. Find the resultant of the four loads and the two points at which the line of action of the resultant intersects the edge of the plate.

Sanu Kumar

Numerade Educator

Problem 109

Gear $C$ is rigidly attached to $\operatorname{arm} A B .$ If the forces and couple shown can be reduced to a single equivalent force at $A,$ determine the equivalent force and the magnitude of the couple $\mathbf{M}$.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 110

To test the strength of a $625 \times 500$ -mm suitcase, forces are applied as shown. If $P=88 \mathrm{~N},(a)$ determine the resultant of the applied forces, $(b)$ locate the two points where the line of action of the resultant intersects the edge of the suitcase.

Dominador Tan

Numerade Educator

Problem 111

Two cables exert forces of $90 \mathrm{kN}$ each on a truss of weight $W=$ $200 \mathrm{kN}$. Find the resultant of these three forces acting on the truss and the point of intersection of its line of action with line $A B$.

Anthony Ramos

Numerade Educator

Problem 112

Pulleys $A$ and $B$ are mounted on bracket $C D E F$. The tension on each side of the two belts is as shown. Replace the four forces with a single equivalent force, and determine where its line of action intersects the bottom edge of the bracket.

Sanu Kumar

Numerade Educator

Problem 113

The roof of a building frame is subjected to the wind loading shown. Determine $(a)$ the equivalent force-couple system at $D,(b)$ the resultant of the loading and its line of action.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 114

A couple of magnitude $M=80 \mathrm{lb} \cdot \mathrm{in} .$ and the three forces shown are applied to an angle bracket. (a) Find the resultant of this system of forces. ( $b$ ) Locate the points where the line of action of the resultant intersects line $A B$ and line $B C$.

Sanu Kumar

Numerade Educator

Problem 115

A couple $\mathbf{M}$ and the three forces shown are applied to an angle bracket. Find the moment of the couple if the line of action of the resultant of the force system is to pass through $(a)$ point $A$, (b) point $B,(c)$ point $C$.

Sanu Kumar

Numerade Educator

Problem 116

A machine component is subjected to the forces and couples shown. The component is to be held in place by a single rivet that can resist a force but not a couple. For $P=0$, determine the location of the rivet hole if it is to be located $(a)$ on line $F G,(b)$ on line $G H$.

Sanu Kumar

Numerade Educator

Problem 117

Solve Prob. 3.116 , assuming that $P=60 \mathrm{~N}$.

Sanu Kumar

Numerade Educator

Problem 118

As follower $A B$ rolls along the surface of member $C,$ it exerts a constant force $\mathbf{F}$ perpendicular to the surface. $(a)$ Replace $\mathbf{F}$ with an equivalent force-couple system at the point $D$ obtained by drawing the perpendicular from the point of contact to the $x$ axis. (b) For $a=1 \mathrm{~m}$ and $b=2 \mathrm{~m}$, determine the value of $x$ for which the moment of the equivalent force-couple system at $D$ is maximum.

Vidhi Bhatt

Numerade Educator

Problem 119

A machine component is subjected to the forces shown, each of which is parallel to one of the coordinate axes. Replace these forces with an equivalent force-couple system at $A$.

Sanu Kumar

Numerade Educator

Problem 120

Two 150 -mm-diameter pulleys are mounted on line shaft $A D$. The belts at $B$ and $C$ lie in vertical planes parallel to the $y z$ plane. Replace the belt forces shown with an equivalent force-couple system at $A$.

Sanu Kumar

Numerade Educator

Problem 121

As an adjustable brace $B C$ is used to bring a wall into plumb, the force-couple system shown is exerted on the wall. Replace this forcecouple system with an equivalent force-couple system at $A$ if $R=$ $21.2 \mathrm{lb}$ and $M=13.25 \mathrm{lb} \cdot \mathrm{ft}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 122

In order to unscrew the tapped faucet $A,$ a plumber uses two pipe wrenches as shown. By exerting a 40 -lb force on each wrench at a distance of 10 in. from the axis of the pipe and in a direction perpendicular to the pipe and to the wrench, he prevents the pipe from rotating, and thus he avoids loosening or further tightening the joint between the pipe and the tapped elbow $C$. Determine $(a)$ the angle $\theta$ that the wrench at $A$ should form with the vertical if elbow $C$ is not to rotate about the vertical, $(b)$ the force-couple system at $C$ equivalent to the two 40 -lb forces when this condition is satisfied.

Sanu Kumar

Numerade Educator

Problem 123

Assuming $\theta=60^{\circ}$ in Prob. 3.122 , replace the two 40 -lb forces with an equivalent force-couple system at $D$ and determine whether the plumber's action tends to tighten or loosen the joint between $(a)$ pipe $C D$ and elbow $D,(b)$ elbow $D$ and pipe $D E$. Assume all threads to be right-handed.

Sanu Kumar

Numerade Educator

Problem 124

Four forces are applied to the machine component $A B D E$ as shown. Replace these forces with an equivalent force-couple system at $A$.

Sanu Kumar

Numerade Educator

Problem 125

A blade held in a brace is used to tighten a screw at $A$. (a) Determine the forces exerted at $B$ and $C$, knowing that these forces are equivalent to a force-couple system at $A$ consisting of $\mathbf{R}=-(25 \mathrm{~N}) \mathbf{i}+R_{y} \mathbf{j}+R_{z} \mathbf{k}$ and $\mathbf{M}_{A}^{R}=-(13.5 \mathrm{~N} \cdot \mathrm{m}) \mathbf{i} .(b)$ Find the
corresponding values of $R_{y}$ and $R_{z}$. $(c)$ What is the orientation of the slot in the head of the screw for which the blade is least likely to slip when the brace is in the position shown?

Sanu Kumar

Numerade Educator

Problem 126

A mechanic uses a crowfoot wrench to loosen a bolt at $C$. The mechanic holds the socket wrench handle at points $A$ and $B$ and applies forces at these points. Knowing that these forces are equivalent to a forcecouple system at $C$ consisting of the force $\mathbf{C}=-(8$ lb $) \mathbf{i}+(4$ lb $) \mathbf{k}$ and the couple $\mathbf{M}_{C}=(360 \mathrm{lb} \cdot \mathrm{in} .) \mathbf{i},$ determine the forces applied at $A$ and at $B$ when $A_{z}=2$ lb.

Vidhi Bhatt

Numerade Educator

Problem 127

Four horizontal forces act on a vertical quarter-circular plate of radius $250 \mathrm{~mm} .$ Determine the magnitude and point of application of the resultant of the four forces if $P=40 \mathrm{~N}$.

Prabhu Ramji

Numerade Educator

Problem 128

Determine the magnitude of the force $P$ for which the resultant of the four forces acts on the rim of the plate.

Ajay Singhal

Numerade Educator

Problem 129

Four signs are mounted on a frame spanning a highway, and the magnitudes of the horizontal wind forces acting on the signs are as shown. Determine the magnitude and the point of application of the resultant of the four wind forces when $a=1 \mathrm{ft}$ and $b=12 \mathrm{ft}$.

Sanu Kumar

Numerade Educator

Problem 130

Four signs are mounted on a frame spanning a highway, and the magnitudes of the horizontal wind forces acting on the signs are as shown. Determine $a$ and $b$ so that the point of application of the resultant of the four forces is at $G$.

Sanu Kumar

Numerade Educator

Problem 131

A concrete foundation mat of 5 -m radius supports four equally spaced columns, each of which is located $4 \mathrm{~m}$ from the center of the mat. Determine the magnitude and the point of application of the resultant of the four loads.

Sanu Kumar

Numerade Educator

Problem 132

Determine the magnitude and the point of application of the smallest additional load that must be applied to the foundation mat of Prob. 3.131 if the resultant of the five loads is to pass through the center of the mat.

Vidhi Bhatt

Numerade Educator

Problem 133

Three forces of the same magnitude $P$ act on a cube of side $a$ as shown. Replace the three forces with an equivalent wrench and determine $(a)$ the magnitude and direction of the resultant force $\mathbf{R},(b)$ the pitch of the wrench, $(c)$ the axis of the wrench.

Sanu Kumar

Numerade Educator

Problem 134

A piece of sheet metal is bent into the shape shown and is acted upon by three forces. If the forces have the same magnitude $P$, replace them with an equivalent wrench and determine $(a)$ the magnitude and the direction of the resultant force $\mathbf{R},(b)$ the pitch of the wrench, $(c)$ the axis of the wrench.

Sanu Kumar

Numerade Educator

Problem 135

The forces and couples shown are applied to two screws as a piece of sheet metal is fastened to a block of wood. Reduce the forces and the couples to an equivalent wrench and determine $(a)$ the resultant force $\mathbf{R},(b)$ the pitch of the wrench, $(c)$ the point where the axis of the wrench intersects the $x z$ plane.

Sanu Kumar

Numerade Educator

Problem 136

The forces and couples shown are applied to two screws as a piece of sheet metal is fastened to a block of wood. Reduce the forces and the couples to an equivalent wrench and determine $(a)$ the resultant force $\mathbf{R},(b)$ the pitch of the wrench, $(c)$ the point where the axis of the wrench intersects the $x z$ plane.

Sanu Kumar

Numerade Educator

Problem 137

Two bolts at $A$ and $B$ are tightened by applying the forces and couples shown. Replace the two wrenches with a single equivalent wrench and determine $(a)$ the resultant $\mathbf{R},(b)$ the pitch of the single equivalent wrench, $(c)$ the point where the axis of the wrench intersects the $x z$ plane.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 138

Two bolts at $A$ and $B$ are tightened by applying the forces and couples shown. Replace the two wrenches with a single equivalent wrench and determine $(a)$ the resultant $\mathbf{R},(b)$ the pitch of the single equivalent wrench, $(c)$ the point where the axis of the wrench intersects the $x z$ plane.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 139

Two ropes attached at $A$ and $B$ are used to move the trunk of a fallen tree. Replace the forces exerted by the ropes with an equivalent wrench and determine $(a)$ the resultant force $\mathbf{R},(b)$ the pitch of the wrench, $(c)$ the point where the axis of the wrench intersects the $y z$ plane.

Vidhi Bhatt

Numerade Educator

Problem 140

A flagpole is guyed by three cables. If the tensions in the cables have the same magnitude $P$, replace the forces exerted on the pole with an equivalent wrench and determine $(a)$ the resultant force $\mathbf{R},(b)$ the pitch of the wrench, $(c)$ the point where the axis of the wrench intersects the $x z$ plane.

Vidhi Bhatt

Numerade Educator

Problem 141

Determine whether the force-and-couple system shown can be reduced to a single equivalent force $\mathbf{R}$. If it can, determine $\mathbf{R}$ and the point where the line of action of $\mathbf{R}$ intersects the $y z$ plane. If it cannot be reduced, replace the given system with an equivalent wrench and determine its resultant, its pitch, and the point where its axis intersects the $y z$ plane

Vidhi Bhatt

Numerade Educator

Problem 142

Determine whether the force-and-couple system shown can be reduced to a single equivalent force $\mathbf{R}$. If it can, determine $\mathbf{R}$ and the point where the line of action of $\mathbf{R}$ intersects the $y z$ plane. If it cannot be reduced, replace the given system with an equivalent wrench and determine its resultant, its pitch, and the point where its axis intersects the $y z$ plane.

Vidhi Bhatt

Numerade Educator

Problem 143

Replace the wrench shown with an equivalent system consisting of two forces perpendicular to the $y$ axis and applied respectively at $A$ and $B$.

Sanu Kumar

Numerade Educator

Problem 144

Show that, in general, a wrench can be replaced with two forces chosen in such a way that one force passes through a given point while the other force lies in a given plane.

Vidhi Bhatt

Numerade Educator

Problem 145

Show that a wrench can be replaced with two perpendicular forces, one of which is applied at a given point.

Vidhi Bhatt

Numerade Educator

Problem 146

Show that a wrench can be replaced with two forces, one of which has a prescribed line of action.

Vidhi Bhatt

Numerade Educator