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12Static Equilibrium and Elasticity

CHAPTER OUTLINE

12.1 The Conditions for Equilibrium12.2 More on the Center of Gravity12.3 Examples of Rigid Objects in

Static Equilibrium12.4 Elastic Properties of Solids

ANSWERS TO QUESTIONS

*Q12.1 The force exerts counterclockwise torque about pivot D. The line of action of the force passes through C, so the torque about this axis is zero. In order of increas-ing negative (clockwise) values come the torques about F, E and B essentially together, and A. The answer is then τ

D > τ

C > τ

F > τ

E = τ

B > τ

A

Q12.2 When you bend over, your center of gravity shifts forward. Once your CG is no longer over your feet, gravity contributes to a nonzero net torque on your body and you begin to rotate.

Q12.3 Yes, it can. Consider an object on a spring oscillating back and forth. In the center of the motion both the sum of the torques and the sum of the forces acting on the object are (separately) zero. Again, a meteoroid fl ying freely through interstellar space feels essentially no forces and keeps moving with constant velocity.

Q12.4 (a) Consider pushing up with one hand on one side of a steering wheel and pulling down equally hard with the other hand on the other side. A pair of equal-magnitude oppositely-directed forces applied at different points is called a couple.

(b) An object in free fall has a non-zero net force acting on it, but a net torque of zero about its center of mass.

*Q12.5 Answer (a). Our theory of rotational motion does not contradict our previous theory of transla-tional motion. The center of mass of the object moves as if the object were a particle, with all of the forces applied there. This is true whether the object is starting to rotate or not.

Q12.6 A V-shaped boomerang, a barstool, an empty coffee cup, a satellite dish, and a curving plastic slide at the edge of a swimming pool each have a center of mass that is not within the bulk of the object.

Q12.7 Suspend the plywood from the nail, and hang the plumb bob from the nail. Trace on the plywood along the string of the plumb bob. Now suspend the plywood with the nail through a different point on the plywood, not along the fi rst line you drew. Again hang the plumb bob from the nail and trace along the string. The center of gravity is located halfway through the thickness of the plywood under the intersection of the two lines you drew.

*Q12.8 In cases (a) and (c) the center of gravity is above the base by one-half the height of the can. So (b) is the answer. In this case the center of gravity is above the base by only a bit more than one-quarter of the height of the can.

311

13794_12_ch12_p311-336.indd 31113794_12_ch12_p311-336.indd 311 1/8/07 8:21:33 PM1/8/07 8:21:33 PM

*Q12.9 Answer (b). The skyscraper is about 300 m tall. The gravitational fi eld (acceleration) is weaker at the top by about 900 parts in ten million, by on the order of 10−4 times. The top half of the uniform building is lighter than the bottom half by about (1�2)(10−4) times. Relative to the center of mass at the geometric center, this effect moves the center of gravity down, by about (1�2)(10−4)(150 m) ∼10 mm.

Q12.10 She can be correct. If the dog stands on a relatively thick scale, the dog’s legs on the ground might support more of its weight than its legs on the scale. She can check for and if necessary correct for this error by having the dog stand like a bridge with two legs on the scale and two ona book of equal thickness—a physics textbook is a good choice.

*Q12.11 Answer (b). Visualize the hatchet as like a balanced playground seesaw with one large-mass person on one side, close to the fulcrum, and a small-mass person far from the fulcrum on the other side. Different masses are on the two sides of the center of mass. The mean position of mass is not the median position.

Q12.12 The free body diagram demonstrates that it is necessary to have friction on the ground to counterbalance the normal force of the wall and to keep the base of the ladder from sliding. If there is friction on the fl oor and on the wall, it is not possible to determine whether the ladder will slip, from the equilibrium conditions alone.

*Q12.13 Answer (g). In the problems we study, the forces applied to the object lie in a plane, and the axis we choose is a line perpendicular to this plane, so it appears as a point on the free-body diagram. It can be chosen anywhere. The algebra of solving for unknown forces is generally easier if we choose the axis where some unknown forces are acting.

*Q12.14 (i) Answer (b). The extension is directly proportional to the original dimension, according to F�A = YΔL�L

i.

(ii) Answer (e). Doubling the diameter quadruples the area to make the extension four times smaller.

Q12.15 Shear deformation.

312 Chapter 12

FIG. Q12.12


Static Equilibrium and Elasticity 313

Section 12.1 The Conditions for Equilibrium

P12.1 Take torques about P.

τ p bn d m g d m gd m gx∑ = − +⎡⎣⎢

⎤⎦⎥

+ +⎡⎣⎢

⎤⎦⎥

+ − =0 1 22 20

� �

We want to fi nd x for which n0 0= .

xm g m g d m g

m g

m m d m

mb b=

+( ) +=

+( ) +1 1 2

2

1 1 2

2

� �

P12.2 Use distances, angles, and forces as shown. The conditionsof equilibrium are:

F F R F

F F R

F F

y y y g

x x x

y

∑∑

∑

= ⇒ + − =

= ⇒ − =

= ⇒ −

0 0

0 0

0τ θ�cos gg xF�

�2

0⎛⎝⎜

⎞⎠⎟ − =cos sinθ θ

Section 12.2 More on the Center of Gravity

P12.3 The coordinates of the center of gravity of piece 1 are

x1 2 00= . cm and y1 9 00= . cm

The coordinates for piece 2 are

x2 8 00= . cm and y2 2 00= . cm

The area of each piece is

A1 72 0= . cm2 and A2 32 0= . cm2

And the mass of each piece is proportional to the area. Thus,

xm x

mi i

iCG

2 2cm cm cm= =

( )( ) + ( )∑∑

72 0 2 00 32 0. . . 88 00

72 0 32 03 85

.

. ..

cm

cm cmcm2 2

( )+

=

and

ym y

mi i

iCG

2 2cm cm cm= =

( )( ) + ( )∑∑

72 0 9 00 32 0. . . 22 00

1046 85

..

cm

cmcm2

( )=

SOLUTIONS TO PROBLEMS

FIG. P12.2

l

Fy

Fx

Ry

Rx

O

Fg

FIG. P12.3

4.00 cm

18.0 cm

12.0 cm

4.00 cm

1

2

FIG. P12.1

x

CG

nOnP

O

m1m2P

d

m g1 m g2m gb

2


314 Chapter 12

P12.4 The hole we can count as negative mass

xm x m x

m mCG = −−

1 1 2 2

1 2

Call σ the mass of each unit of pizza area.

xR

R

xR

R R

RCG

CG

=− ( ) −( )

− ( )

=

σπ σπσπ σπ

22

22

22

2

0

8

3

/

/44 6= R

P12.5 Let the fourth mass (8.00 kg) be placed at (x, y), then

xm x

m

x

CG = = ( )( ) + ( )+

= − =

03 00 4 00

12 0

12 0

8 00

4

4

. .

.

.

.−−1 50. m

Similarly,

yy

CG = =( )( ) + ( )

+0

3 00 4 00 8 00

12 0 8 00

. . .

. .

y = −1 50. m

P12.6 Let σ represent the mass-per-face area. A vertical strip at position x, with width

dx and height x −( )3 00

9

2. has mass

dmx dx= −( )σ 3 00

9

2.

The total mass is

M dmx dx

M x x

x

= = −( )

= ⎛⎝⎜

⎞⎠⎟ − +( )

∫ ∫=

σ

σ

3

9

96 9

2

0

3 00

2

.

ddx

Mx x

x

0

3 00

3 2

0

3 00

9 3

6

29

.

.

∫

= ⎛⎝⎜

⎞⎠⎟ − +⎡

⎣⎢⎤⎦⎥

=σ σ

The x-coordinate of the center of gravity is

xxdm

Mx x dx x x xCG = = −( ) = − +(∫ ∫

1

93

96 92

0

3 003 2

σσ σ

σ

.

))

= − +⎡⎣⎢

⎤⎦⎥

=

∫ dx

x x x

0

3 00

4 3 2

0

3 001

9 4

6

3

9

2

6 75

.

.. mm

9.00m= 0 750.

FIG. P12.6

x

dx0

3.00 m

x

1.00 m

y = (x – 3.00)2/9

y



P12.7 In a uniform gravitational fi eld, the center of mass and center of gravity of an object coincide. Thus, the center of gravity of the triangle is located at x = 6 67. m, y = 2 33. m (see the Example on the center of mass of a triangle in Chapter 9).

The coordinates of the center of gravity of the three-object system are then:

xm x

mi i

iCG

kg m kg= = ( )( ) + ( )∑

∑6 00 5 50 3 00 6 6. . . . 77 5 00 3 50

6 00 3 00 5 00

m kg m

k

( ) + ( ) −( )+ +( )

. .

. . . gg

kg m

14.0 kgm andCG

CG

x

ym y

mi i

= ⋅ =

= ∑

35 52 54

..

ii∑= ( )( ) + ( )( ) +6 00 7 00 3 00 2 33 5 0. . . . .kg m kg m 00 3 50

14 0

66 5

kg m

kg

kg m

14.0 kgCG

( ) +( )

= ⋅

.

.

.y == 4 75. m

Section 12.3 Examples of Rigid Objects in Static Equilibrium

P12.8 (a) For rotational equilibrium of the lowest rod about its point of support, τ∑ = 0.

+ −12 0 1. g 3 cm 4 cmg m g m1 9 00= . g

(b) For the middle rod,

+ − +( ) =m2 2 12 0 9 0 5 0cm g g cm. . m2 52 5= . g

(c) For the top rod,

52 5 12 0 9 0 4 6 03. . .g g g cm cm+ +( ) − =m m3 49 0= . g

P12.9 τ∑ = = ( ) −0 3mg r Tr

2 45 0 0

45 0

2

1 500 4

T Mg

TMg g

− =

= =( )

sin .

sin . sin

°

° kg 55 0

2530 9 80

3

530

3177

.

.

°

= ( )( )

= = =

N

kgmT

g

g

g

P12.10 (a) Taking moments about P,

R Rsin . cos . .30 0 0 30 0 5 00 150 30° °( ) + ( )( ) − ( )cm N ..

. .

0 0

1 039 2 1 04

cm

N kN

( ) == =R

The force exerted by the hammer on the nail is equal in magnitude and opposite in direction:

1 04. kN at 60 upward and to the right.°

(b) f R= − =sin .30 0 150 370° N N

n R= =

= ( ) + (

cos .

ˆ

30 0 900

370 900

° N

N Nsurface

�F i )) j

FIG. P12.9

m

3r

θ = 45°

1 500 kg


316 Chapter 12

P12.11 (a) F f nx w∑ = − = 0

F ny g∑ = − − =800 500 0N N

Taking torques about an axis at the foot of the ladder,

800 4 00 30 0 500 7 50 3N m N m( )( ) + ( )( ). sin . . sin° 00 0. °

15 0 30 0 0. cos . °− ( ) =nw cm

Solving the torque equation,

nw =( )( ) + ( )( )[ ]4 00 800 7 50 500 30 0. . tan .m N m N °°

15 0268

. mN=

Next substitute this value into the Fx equation to fi nd

f nw= = 268 N in the positive x direction.

Solving the equation Fy∑ = 0,

ng = 1 300 N in the positive y direction.

(b) In this case, the torque equation τ A =∑ 0 gives:

9 00 800 30 0 7 50 500 3. sin . . sinm N m N( )( ) + ( )( )° 00 0 15 0 60 0 0. . sin .° °− ( )( ) =m nw

or

nw = 421 N

Since f nw= = 421 N and f f ng= =max μ , we fi nd

μ = = =f

ng

max .421

0 324N

1 300 N

P12.12 (a) F f nx w∑ = − = 0 (1)

F n m g m gy g∑ = − − =1 2 0 (2)

τ θ θ θA wm gL

m gx n L∑ = − ⎛⎝⎜

⎞⎠⎟ − + =1 22

0cos cos sin

From the torque equation,

n m gx

Lm gw = + ⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

1

2 1 2 cotθ

Then, from equation (1): f n m gx

Lm gw= = + ⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

1

2 1 2 cotθ

and from equation (2): n m m gg = +( )1 2

(b) If the ladder is on the verge of slipping when x d= ,

then

μθ

= =+( )

+=f

n

m m d L

m mx d

g

1 2

1 2

2/ / cot

FIG. P12.12

f

A n g

m g1

m g2

n w

θ

FIG. P12.11

ng

f

nw

500 N

800 N

A



P12.13 Torque about the front wheel is zero.

0 1 20 3 00 2= ( )( ) − ( )( ). .m mmg Fr

Thus, the force at each rear wheel is

F mgr = =0 200 2 94. . kN

The force at each front wheel is then

F

mg Ff

r= − =2

24 41. kN

*P12.14 (a) The gravitational force on the fl oodlight is (20 kg)(9.8 m�s2) = 196 N We consider the torques acting on the beam,

about an axis perpendicular to the page and through the left end of the horizontal beam.

τ∑ = +( ) − ( ) =T d dsin .30 0 196 0° N

giving T = 392 N .

(b) From Fx =∑ 0, H T− =cos .30 0 0° , or H = ( ) =392 30 0 339N N to the rightcos . ° .

(c) From Fy∑ = 0,V T+ − =sin .30 0 196 0° N , orV = − ( ) =196 392 30 0 0N N sin . ° .

(d) From the same free-body diagram with the axis chosen at the right-hand end, we write

∑τ = H(0) − Vd + T(0) + 196 N(0) = 0 so V = 0

(e) From Fy∑ = 0,V T+ − =sin .30 0 196 0° N , or T = + =0 196 30 0 392N/s Nin . ° .

(f ) From Fx =∑ 0, H T− =cos .30 0 0° , or H = ( ) =392 30 0 339N N to the rightcos . ° .

(g) The two solutions agree precisely. They are equally accurate. They are essentially equally simple. But note that many students would make a mistake on the negative (clockwise) sign for the torque of the upward force V in the equation in part (d).

Taking together the equations we have written, we appear to have four equations but we cannot determine four unknowns. Only three of the equations are independent, so we can determine only three unknowns.

P12.15 (a) Vertical forces on one half of the chain: Te sin . .42 0 20 0° = N Te = 29 9. N

(b) Horizontal forces on one half of the chain: T Te mcos .42 0° = Tm = 22 2. N

FIG. P12.13

FIG. P12.14

H

d196 N

V

T

30.0°


318 Chapter 12

P12.16 Relative to the hinge end of the bridge, the cable is attached

horizontally out a distance x = ( ) =5 00 20 0 4 70. cos . .m m°

and vertically down a distance y = ( ) =5 00 20 0. sin .m °

1 71. m. The cable then makes the following angle with the horizontal:

θ = +( )⎡⎣⎢

⎤⎦⎥

=−tan. .

.1 12 0 1 7171 1

m

4.70 m°

(a) Take torques about the hinge end of the bridge:

R R

T

x y0 0 19 6 20 0

71

( ) + ( ) − ( )−

. cos .

cos .

kN 4.00 m °

11 1 71 71 1 4 70

9 80

° °. sin . .

.

m m

kN 7.00 m

( ) + ( )−

T

(( ) =cos .20 0 0°

which yields T = 35 5. kN

(b) F R Tx x∑ = ⇒ − =0 71 1 0cos . °

or

Rx = ( ) = ( )35 5 71 1 11 5. cos . .kN kN right°

(c) F R Ty y∑ = ⇒ − + − =0 19 6 71 1 9 80 0. sin . .kN kN°

Thus,

Ry = − ( ) = −

=

29 4 35 5 71 1 4 19

4 19

. . sin . .

.

kN kN kN°

kkN down

P12.17 (a) We model the horse as a particle. The drawbridge will fall out from under the horse.

α θ θ= =

=( )

mgm

g12 0

13

2 0

3

2

3 9 80 2

�

� �

coscos

. cosm s2 00 0

2 8 001 73

.

..

°

mrad s2

( )=

(b) 1

22I mghω =

∴ ⋅ = ⋅ −( )1

2

1

3

1

212 2

0m mg� �ω θsin

Solving,

m s

m

2

ω θ= −( ) =( )

−31

3 9 80

8 0010

g

�sin

.

.ssin .20 1 56°( ) = rad s

continued on next page

FIG. P12.17(a)

RyRx

mg

θ0

FIG. P12.16

x

y

T

20.0°Rx

Ry

9.80 kN

19.6 kN

4.00 m

5.00 m

7.00 m



(c) The linear acceleration of the center of mass of the bridge is

a = = ( )( ) =1

2

1

28 0 1 73 6 907�α . . .m rad s m s2 2

The force at the hinge plus the gravitational force produce the acceleration a = 6 907. m s2

at right angles to the bridge.

R max x= = ( )( ) = −2 000 6 907 250 4 72kg m s kN2. cos .°

R mg may y− =

Solving,

kg m s2R m g ay y= +( ) = ( ) +2 000 9 80 6 90. . 77 250 6 62m s kN2( )⎡⎣ ⎤⎦ =sin .°

Thus�R i j= − +( )4 72 6 62. ˆ . ˆ kN

(d) Rx = 0

a

R

= ⎛⎝⎜

⎞⎠⎟ = ( ) ( ) =ω 2 21

21 56 4 0 9 67� . . .rad s m m s2

yy

y

mg ma

R

− =

∴ = ( ) +( ) =2 000 9 8 9 67 38kg m s m s2 2. . .99 kN

Thus:

Ry = 38 9. j kN

P12.18 Call the required force F, with components F Fx = cos .15 0° andF Fy = − sin .15 0°, transmitted to the center of the wheel by the handles.

Just as the wheel leaves the ground, the ground exerts no force on it.

Fx∑ = 0: F nxcos .15 0° − (1)

Fy∑ = 0: − − + =F nysin .15 0 400 0° N (2)

Take torques about its contact point with the brick. The needed distances are seen to be:

b R

a R b

= − = −( ) =

= − =

8 00 20 0 8 00 12 0

12 2

. . . .cm cm cm

66 0. cm

(a) τ∑ = 0: − + + ( ) =F b F a ax y 400 0N , or

F −( ) + ( )[ ]+12 0 15 0 16 0 15 0 400. cos . . sin .cm cm° ° N cm( )( ) =16 0 0.

so

F = ⋅ =6 400859

N cm

7.45 cmN

FIG. P12.17(d)

RyRx

mg

a

FIG. P12.18

RRR

nxny

Fx

Fy 400 N

b

8.00 cm

distances forces

aa

b

a

FIG. P12.17(c)

RyRx

mg

θ0

a



320 Chapter 12

(b) Then, using Equations (1) and (2),

nx = ( ) =859 15 0 830N Ncos . ° and

ny = + ( ) =400 859 15 0 622N N Nsin . °

n n n

n

n

x y

y

x

= + =

=⎛⎝⎜

⎞⎠⎟

=− −

2 2

1 1

1 04

0 7

.

tan tan .

kN

θ 449 36 9( ) = . ° to the left and upward

P12.19 When x x= min, the rod is on the verge of slipping, so

f f n ns s= ( ) = =max

.μ 0 50

From Fx∑ = 0, n T− =cos 37 0° , or n T= 0 799. .

Thus,

f T T= ( ) =0 50 0 799 0 399. . .

From Fy∑ = 0, f T Fg+ − =sin 37 2 0° ,

or

0 399 0 602 2 0. .T T Fg− − = , giving T Fg= 2 00.

Using τ∑ = 0 for an axis perpendicular to the page and through the left end of the beam gives

− ⋅ − ( ) + ( )⎡⎣ ⎤⎦( ) =F x F Fg g gmin . sin .2 0 2 37 4 0 0m m° , which reduces to xmin .= 2 82 m

P12.20 Consider forces and torques on the beam.

Fx∑ = 0: R Tcos cosθ − =53 0°

Fy∑ = 0: R Tsin sinθ + − =53 800 0° N

τ∑ = 0: T xsin 53 8 600 200 4 0°( ) − ( ) − ( ) =m N N m

(a) Then Tx

x= + ⋅ = ( ) +600 800

5393 9 125

N N m

8 mN m N

sin.

°

As x increases from 2 m, this expression grows larger.

(b) From substituting back,

R x

R x

cos . cos

sin .

θθ

= +[ ]= − +

93 9 125 53

800 93 9 12

°

N 55 53[ ]sin °

Dividing,

tansin

costan

cosθ θ

θ= = − +

( )R

R x53

800

93 9 125°

N

. + 553°

tan tanθ =+

−⎛⎝⎜

⎞⎠⎟53

32

3 41°

x

As x increases the fraction decreases and θ decreases .

FIG. P12.19

37°

xf

n

F

T

g Fg

2.0 m

2.0 m




(c) To fi nd R we can work out R R R2 2 2 2 2cos sinθ θ+ = . From the expressions above for R cosθ and R sinθ ,

R T T T2 2 2 2 253 53 1 600 53 800= + − + (cos sin sin° ° °N N))= − +

= +( ) −

2

2 2

2 2

1 600 53 640 000

93 9 125

R T T

R x

sin

.

°

11 278 93 9 125 640 000

8 819 96 482 495 62

. x

R x x

+( ) +

= − + 7781 2( )

At x = 0 this gives R = 704 N. At x = 2 m, R = 581 N. At x = 8 m, R = 537 N. Over the range of possible values for x, the negative term −96 482x dominates the positive term

8 819 2x , and R decreases as x increases.

P12.21 To fi nd U, measure distances and forces from point A. Then, balancing torques,

0 750 29 4 2 25. . .( ) = ( )U U = 88 2. N

To fi nd D, measure distances and forces from point B. Then, balancing torques,

0 750 1 50 29 4. . .( ) = ( )( )D D = 58 8. N

Also, notice that U D Fg= + , so Fy =∑ 0.

Section 12.4 Elastic Properties of Solids

P12.22 The defi nition of Y = stress

strain means that Y is the slope of the graph:

Y = × = ×300 10

0 0031 0 10

611N m

N m2

2

..

P12.23 F

A

L

Li

= YΔ

ΔLFL

Ai= = ( )( )( )

×( ) ×−Y

200 9 80 4 00

0 200 10 8 004

. .

. . 1104 9010( ) = . mm

P12.24 (a) stress = =F

A

F

rπ 2

F

d

F

= ( ) ⎛⎝

⎞⎠

= ×( ) ×

stress

N m2.50 102

π

π

2

1 50 10

2

8.−22 m

2

kN

⎛⎝⎜

⎞⎠⎟

=

2

73 6F .

(b) stress = ( ) =YY L

Li

strainΔ

ΔLL

Yi= ( )

=×( )( )

×stress N m m21 50 10 0 250

1 50

8. .

. 1102 5010 N m

mm2 = .


322 Chapter 12

P12.25 From the defi ning equation for the shear modulus, we fi nd Δx as

Δxhf

SA= =

×( )( )×( )

−5 00 10 20 0

3 0 10 1

3

6

. .

.

m N

N m2 44 0 102 38 104

5

..

×( ) = ×−−

mm2

or Δx = × −2 38 10 2. mm

P12.26 Count the wires. If they are wrapped together so that all support nearly equal stress, the number should be

20 0100

. kN

0.200 kN=

Since cross-sectional area is proportional to diameter squared, the diameter of the cable will be

1 100 1mm cm( ) ~

P12.27 (a) F A= ( )( )= ×( ) ×( )−

stress

m N m2π 5 00 10 4 00 103 2 8. .

== 3.14 10 N4×

(b) The area over which the shear occurs is equal to the circumference of the hole times its thickness. Thus,

A r t= ( ) = ×( ) ×( )

= ×

− −2 2 5 00 10 5 00 10

1 57

3 3π π . .

.

m m

110 4− m2

So,

F A= ( ) = ×( ) ×( )−Stress m N m2 21 57 10 4 00 104 8. .

= 6.228 104× N

P12.28 The force acting on the hammer changes its momentum according to

m F t mi fv v+ ( ) =Δ so Fm

tf i=

−v v

Δ

Hence,

F =− −

= ×30 0 10 0 20 0

0 1108 18 103. . .

..

kg m s m s

sN

By Newton’s third law, this is also the magnitude of the average force exerted on the spike by the hammer during the blow. Thus, the stress in the spike is:

stress = = ×( )

= ×F

A

8 18 10

41 97 10

3

27.

/.

N

0.023 0 mN m2

π

and the strain is: strain = = ××

= × −stress N m

N m

2

2Y

1 97 10

20 0 109 85 10

7

10

.

.. 55

FIG. P12.27(b)

F

3.0 ft

t

AAA



P12.29 Consider recompressing the ice, which has a volume 1 09 0. V .

Δ ΔP B

V

Vi

= −⎛⎝⎜

⎞⎠⎟

=− ×( ) −( )2 00 10 0 090

1 09

9. .

.

N m2

== ×1 65 108. N m2

P12.30 Let the 3.00 kg mass be mass #1, with the 5.00 kg mass, mass # 2. Applying Newton’s second law to each mass gives:

m a T m g1 1= − (1) and m a m g T2 2= − (2)

where T is the tension in the wire.

Solving equation (1) for the acceleration gives: aT

mg= −

1

and substituting this into equation (2) yields: m

mT m g m g T2

12 2− = −

Solving for the tension T gives

Tm m g

m m=

+=

( )( )( )2 2 3 00 5 00 9 801 2

2 1

. . .kg kg m s2

88 0036 8

..

kgN=

From the defi nition of Young’s modulus, YFL

A Li=

( )Δ, the elongation of the wire is:

ΔLTL

YAi= = ( )( )

×( )36 8 2 00

2 00 10 211

. .

. .

N m

N m2 π 000 100 029 3

3 2×( ) =

− mmm.

P12.31 Part of the load force extends the cable and part compresses the column by the same distance Δ�:

F

Y A Y A

F

Y A Y A

A A

A

s s

s

A A A s s s

= +

=+

=

Δ Δ

Δ

�

�

�

�

�� / /

8 500 N

7 1010× −( ) ( ) + ×π 0 162 4 0 161 4 4 3 25 20 102 2 10. . / . ππ 0 012 7 4 5 75

8 60 10

2

4

. / .

.

( ) ( )

= × − m

P12.32 BP

V V

PV

Vi

i= − = −ΔΔ

ΔΔ/

(a) Δ ΔV

PV

Bi= − = −

×( )×

=1 13 10 1

0 21 10

8

10

.

.

N m m

N m

2 3

2 −−0 053 8. m3

(b) The quantity of water with mass 1 03 103. × kg occupies volume at the bottom

1 0 053 8 0 946m m m3 3 3− =. . . So its density is 1 03 101 09 10

33.

.× = ×kg

0.946 mkg m3

3

(c) With only a 5% volume change in this extreme case, liquid water is indeed nearly

incompressible.


324 Chapter 12

Additional Problems

P12.33 Let nA and nB be the normal forces at the points of support.

Choosing the origin at point A with Fy∑ = 0 and τ∑ = 0, we fi nd:

n n g gA B+ − ×( ) − ×( ) =8 00 10 3 00 10 04 4. . and

− ×( )( ) − ×( )( ) +3 00 10 15 0 8 00 10 25 0 50 04 4. . . . .g g nB (( ) = 0

The equations combine to give nA = ×5 98 105. N and bB = ×4 80 105. N .

*P12.34 Model the stove as a uniform 68 kg box. Its center of mass is at its

geometric center, 28

214= inches behind its feet at the front corners.

Assume that the light oven door opens to be horizontal and that aperson stands on its outer end, 46 375 28 18 375. .− = inches in front of the front feet.

We fi nd the weight Fg of a person standing on the oven door with the stove balanced on its front

feet in equilibrium: τ∑ = 0

68 9 8 14 0 18 375kg m s in. in.2( )( )( ) + ( ) − (. .n Fg )) =

=

0

508Fg N

If the weight of the person is greater than this, the stove can tip forward. This weight corre-sponds to mass 51.8 kg, so the person could be a child. If the oven door is heavy (compared to the backsplash) or if the front feet are signifi cantly far behind the front corners, the maxi-mum weight will be signifi cantly less than 508 N.

P12.35 With � as large as possible, n1 and n2 will both be large. The equality sign in f ns2 2≤ μ will be true, but the less-than sign will apply in f ns1 1< μ . Take torques about the lower end of the pole.

n F fg2 2

1

20� � �cos cos sinθ θ θ+ ⎛

⎝⎞⎠ − =

Setting f n2 20 576= . , the torque equation becomes

n Fg2 1 0 5761

20−( ) + =. tanθ

Since n2 0> , it is necessary that

1 0 576 0

1

0 5761 736

60 1

− <

∴ > =

∴ >

∴ =

. tan

tan.

.

.

θ

θ

θ °

�d

ssin

.

sin ..

θ< =7 80

60 19 00

ftft

°

FIG. P12.35

f1

f2

n1n2 Fg

θ

θ

d

FIG. P12.33

BA

15.0 m15.0 m50.0 m50.0 m

FIG. P12.34

666 NFg

n



P12.36 When the concrete has cured and the pre-stressing tension has been released, the rod presses in on the concrete and with equal force, T2, the concrete produces tension in the rod.

(a) In the concrete: stress = × = ⋅( ) =⎛⎝⎜

⎞⎠⎟

8 00 106. N m strain2 Y YL

Li

Δ

Thus,

ΔLL

Yi= ( )

=×( )( )

×stress N m m28 00 10 1 50

30 0 1

6. .

. 009 N m2

or

ΔL = × =−4 00 10 0 4004. .m mm

(b) In the concrete: stress = = ×T

Ac

2 68 00 10. N m2, so

T26 48 00 10 50 0 10 40 0= ×( ) ×( ) =−. . .N m m kN2 2

(c) For the rod: T

A

L

LY

R i

2 =⎛⎝⎜

⎞⎠⎟

Δsteel so ΔL

T L

A Yi

R

= 2

steel

ΔL =×( )( )

×( ) ×−

4 00 10 1 50

1 50 10 20 0 1

4

4

. .

. .

N m

m2 002 00 10 2 0010

3

N mm mm2( ) = × =−. .

(d) The rod in the fi nished concrete is 2.00 mm longer than its unstretched length. To remove stress from the concrete, one must stretch the rod 0.400 mm farther, by a total of

2 40. mm .

(e) For the stretched rod around which the concrete is poured:

T

A

L

LY T

L

LR i i

11=

⎛⎝⎜

⎞⎠⎟

=⎛⎝⎜

Δ Δtotalsteel

totalor⎞⎞⎠⎟

= ×⎛⎝⎜

⎞⎠⎟

×−

A Y

T

R steel

m

1.50 m1

32 40 101 50

.. 110 20 0 10 48 04 10−( ) ×( ) =m N m kN2 2. .

P12.37 (a) See the diagram.

(b) If x = 1 00. m, then

τO∑ = −( )( ) − ( )( )−

700 1 00 200 3 00

80 0

N m N m

N

. .

.(( )( )+ ( )( ) =

6 00

60 0 6 00 0

.

sin . .

m

mT °

Solving for the tension gives: T = 343 N .

From Fx∑ = 0, R Tx = =cos .60 0 171° N .

From Fy∑ = 0, R Ty = − =980 60 0 683N Nsin . ° .

(c) If T = 900 N:

τ

Ox∑ = −( ) −( )( )−( )700 200 3 0 80 0 6 00N N 0 m N. . . mm

N m

( )+ ( )⎡⎣ ⎤⎦( )=900 60 0 6 00 0sin . .°

Solving for x gives: x = 5 13. m .

FIG. P12.37

Ry

x

3.00 m

O

3.00 m

Rx 60.0°

T

700 N200 N 80.0 N


326 Chapter 12

*P12.38 The 392 N is the weight of the uniform gate, which is 3 mwide. The hinges are 1.8 m apart. They exert horizontal forces A and C. Only one hinge exerts a vertical force.We assume it is the upper hinge.

(a) Free body diagram:

Statement:

A uniform 40.0-kg farm gate, 3.00 m wide and 1.80 m high, supports a 50.0-N bucket of grain hanging from its latch as shown. The gate is supported by hinges at two corners. Find the force each hinge exerts on the gate.

(b) From the torque equation,

C = ⋅ =738410

N m

1.8 mN

Then A = 410 N. Also B = 442 N.

The upper hinge exerts 410 N to the left andd 442 N up.

The lower hinge exerts 410 N to the right.

P12.39 Using F Fx y∑ ∑ ∑= = =τ 0, choosing the origin at the left

end of the beam, we have (neglecting the weight of the beam)

F R T

F R T Fx x

y y g

∑∑

= − =

= + − =

cos

sin

θ

θ

0

0

and τ θ∑ = − +( ) + +( ) =F L d T L dg sin 2 0.

Solving these equations, we fi nd:

(a) TF L d

L dg=

+( )+( )sinθ 2

(b) RF L d

L dxg=

+( )+

cotθ2

RF L

L dyg=+2

P12.40 τ point 0∑ = 0 gives

T Tcos . sin . sin .25 03

465 0 25 0

3

4° ° °( )⎛

⎝⎜⎞⎠⎟ + ( )� �

ccos .

cos .

65 0

2 000 65 0 1 200

°

°

⎛⎝⎜

⎞⎠⎟

= ( )( ) + (N N� ))⎛⎝⎜

⎞⎠⎟

�

265 0cos . °

From which, T = =1 465 1 46N kN.

From Fx∑ = 0,

H T= = ( ) =cos . .25 0 1 328 1 33° N toward right kN

From Fy∑ = 0,

V T= − = ( ) =3 200 25 0 2 581 2 58N N upward kNsin . .°

FIG. P12.39

H

V

65.0°

1 200 N

l 2 000 N

3

4

l

T sin .25 0°

T cos .25 0°

FIG. P12.40

FIG. P12.38

AB

C392 N

50 N

3.0 m

1.8 m



P12.41 We interpret the problem to mean that the support at point Bis frictionless. Then the support exerts a force in thex direction and

F

F F F

F g

By

x Bx Ax

Ay

=

= − =

− +( ) =∑

0

0

3 000 10 000 0

and

τ∑ = −( )( ) − ( )( ) + ( ) =3 000 2 00 10 000 6 00 1 00g g FBx. . . 00

These equations combine to give

F F

F

Ax Bx

Ay

= = ×

= ×

6 47 10

1 27 10

5

5

.

.

N

N

*P12.42 Choosing torques about the hip joint, τ∑ = 0 gives

− ( ) + ( )⎛⎝

⎞⎠ − ( ) =L

TL

L2

350 12 02

3200 0N Nsin . °

From which, T = 2 71. kN .

Let Rx = compression force along spine, and from Fx∑ = 0

R T Tx x= = =cos . .12 0 2 65° kN

(c) You should lift “with your knees” rather than “with your back.” In this situation, with a load weighing only 200 N, you can make the compressional force in your spine about ten times smaller by bending your knees and lifting with your back as straight as possible.

P12.43 From the free-body diagram, the angle that the string tension makes with the rod is

θ = + =60 0 20 0 80 0. . .° ° °

and the perpendicular component of the string tension is T sin .80 0°. Summing torques around the base of the rod gives

τ∑ = 0: −( )( ) + ( ) =4 00 10 000 60 4 00 80 0. cos . sinm N m° °T

T =( ) = ×10 000 60 0

80 05 08 103N

Ncos .

sin ..

°

°

Fx∑ = 0: F TH − =cos .20 0 0°

F TH = = ×cos . .20 0 4 77 103° N

Fy∑ = 0: F TV + − =sin .20 0 10 000 0° N

and F TV = ( ) − = ×10 000 20 0 8 26 103N Nsin . .°

FIG. P12.42

FIG. P12.43

FV

T

60°

20°

FH

10 000 N

FIG. P12.41


328 Chapter 12

P12.44 (a) Just three forces act on the rod: forces perpendicular to the sides of the trough at A and B, and its weight. The lines of action of A and B will intersect at a point above the rod. They will have no torque about this point. The rod’s weight will cause a torque about the point of intersection as in Figure 12.52(a), and the rod will not be in equilibrium unless the center of the rod lies vertically below the intersection point, as in Figure 12.52(b). All three forces must be concur-rent. Then the line of action of the weight is a diagonal of the rectangle formed by the trough and the normal forces, and therod’s center of gravity is vertically above the bottom of the trough.

(b) In Figure (b), AO BOcos . cos .30 0 60 0° °= and

L2 2 2 2 2

2

2

30 0

60 0= + = +

⎛⎝⎜

⎞⎠⎟

AO BO AO AO

A

cos .

cos .

°

°

OO =+ ⎛

⎝⎞⎠

=L L

13060

22coscos

°°

So

cosθ = =AO

L

1

2 and θ = 60 0. °

P12.45 (a) Locate the origin at the bottom left corner of the cabinet and let x = distance between the resultant normal force and the front of the cabinet. Then we have

F nx∑ = − =200 37 0 0cos . ° μ (1)

F ny = + − =∑ 200 37 0 400 0sin . ° (2)

τ∑ = −( ) − ( ) + ( )n x0 600 400 0 300 200 37 0 0 600. . sin . .°

− ( ) =200 37 0 0 400 0cos . .° (3)

From (2),

n = − =400 200 37 0 280sin . ° N

From (3),

x = − + ( ) −72 2 120 280 0 600 64 0

280

. . .

x = 20 1. cm to the left of the front edge

From (1), μk = =200 37 0

2800 571

cos ..

°

(b) In this case, locate the origin x = 0 at the bottom right corner of the cabinet. Since the cabinet is about to tip, we can use τ∑ = 0 to fi nd h:

τ∑ = ( ) − ( ) =400 0 300 300 37 0 0. cos . ° h h = =120

300 37 00 501

cos ..

°m

FIG. P12.45

FIG. P12.44(b)

A

B

Fg

O

θ30.0° 60.0°

FIG. P12.44(a)

A

B

Fg

O



P12.46 (a), (b) Use the fi rst diagram and sum the torques about the lower front corner of the cabinet.

τ∑ = ⇒ − ( ) + ( )( ) =0 1 00 400 0 300 0F . .m N m

yielding F = ( )( )=400 0 300

1 00120

N m

mN

.

.

F fx∑ = ⇒ − + =0 120 0N , or f = 120 N

F ny∑ = ⇒ − + =0 400 0N , so n = 400 N

Thus,

μs

f

n= = =120

0 300N

400 N.

(c) Apply ′F at the upper rear corner and directed so θ φ+ = 90 0. ° to obtain the largest possible lever arm.

θ = ⎛⎝

⎞⎠ =−tan

..1 1 00

59 0m

0.600 m°

Thus, φ = − =90 0 59 0 31 0. . .° ° °

Sum the torques about the lower front corner of the cabinet:

− ′ ( ) + ( ) + ( )( ) =F 1 00 0 600 400 0 300 02 2. . .m m N m

so

′ = ⋅ =F120

103N m

1.17 mN

Therefore, the minimum force required to tip the cabinet is

103 N applied at 31.0 above the horizontal° at the upper left corner .

P12.47 (a) We can use F Fx y∑ ∑= = 0 and τ∑ = 0 with pivot point at the contact on the fl oor.

Then

F T nx s∑ = − =μ 0

F n Mg mgy = − − =∑ 0, and

τ θ θ θ∑ = ( ) + ⎛⎝

⎞⎠ − ( ) =Mg L mg

LT Lcos cos sin

20

Solving the above equations gives

Mm s

s

= −−

⎛⎝⎜

⎞⎠⎟2

2μ θ θθ μ θsin cos

cos sin

This answer is the maximum value for M if μ θs < cot . If μ θs ≥ cot , the mass M can increase without limit. It has no maximum value.

(b) At the fl oor, we have the normal force in the y-direction and frictional force in the x-direction. The reaction force then is

R n n M m gs s= + ( ) = +( ) +2 2 21μ μ

At point P, the force of the beam on the rope is

F T Mg g M M ms= + ( ) = + +( )2 2 2 2 2μ

FIG. P12.46

400 N

nf

F

0.300 m

1.00 m

n

1.00 m

0.600 m

400 N

θ

θ

φF'

f

FIG. P12.47

n

f

P

θ

mg

T

MgL/2

L/2


330 Chapter 12

*P12.48 Suppose that a bar exerts on a pin a force not along the length of the bar. Then, the pin exerts on the bar a force with a component per-pendicular to the bar. The only other force on the bar is the pin force on the other end. For

�F∑ = 0, this force must also have a compo-

nent perpendicular to the bar. Then each of the forces produces a torque about the center of the bar in the same sense. The total torque on the bar is not zero. The contradiction proves that the bar can only exert forces along its length.

FIG. P12.48

*P12.49 (a) The height of pin B is

10 0 30 0 5 00. sin . .m m( ) =°

The length of bar BC is then

BC = =5 00

45 07 07

.

sin ..

mm

°

Consider the entire truss:

F n ny A C

A

∑∑

= − + =

= −( )1 000 0

1 000 10 0 30 0

N

Nτ . cos . ° ++ +[ ] =nC 10 0 30 0 7 07 45 0 0. cos . . cos .° °

Which gives nC = 634 N .

Then,

n nA C= − =1 000 366N N

(b) Joint A:

Fy =∑ 0: − + =CABsin .30 0 366 0° N

so

CAB = 732 N

Fx∑ = 0: − + =C TAB ACcos .30 0 0°

TAC = ( ) =732 30 0 634N Ncos . °

Joint B:

Fx∑ = 0: 732 30 0 45 0 0N( ) − =cos . cos .° °CBC

CBC = ( )=732 30 0

45 0897

NN

cos .

cos .

°

°

FIG. P12.49(b)

CAB

ATAC

nA = 366 N

CBC

B

CAB = 732 N

1000 N

30.0° 45.0°

FIG. P12.49(a)

1000 N

B

A C

10.0 mnA nC

30.0° 45.0°



*P12.50 Considering the torques about the point at the bottom of the bracket yields:

W F0 050 0 0 060 0 0. .m m( ) − ( ) = so F = 0.833W

(a) With W = 80.0 N, F = 0.833(80 N) = 66 7. N .

(b) Differentiate with respect to time: dF�dt = 0.833 dW�dt

The force exerted by the screw is increasing at the rate dF�dt = 0.833(0.15 N�s) = 0.125 N�s

P12.51 From geometry, observe that

cosθ = 1

4 and θ = 75 5. °

For the left half of the ladder, we have

F T Rx x∑ = − = 0 (1)

F R ny y A∑ = + − =686 0N (2)

τ top N∑ = ( ) + ( )686 1 00 75 5 2 00 75 5. cos . . sin .° °T

− ( ) =nA 4 00 75 5 0. cos . ° (3)

For the right half of the ladder we have

F R Tx x∑ = − = 0

F n Ry B y∑ = − = 0 (4)

τ top∑ = ( ) − ( ) =n TB 4 00 75 5 2 00 75 5 0. cos . . sin .° ° (5)

Solving equations 1 through 5 simultaneously yields:

(a) T = 133 N

(b) nA = 429 N and nB = 257 N

(c) Rx = 133 N and Ry = 257 N

The force exerted by the left half of the ladder on the right half is to the right and downward.

FIG. P12.51

T T


332 Chapter 12

P12.52 Imagine gradually increasing the force P. This will make the force of static friction at the bottom increase, so that the normal force at the wall increases and the friction force at the wall can increase. As P reaches its maximum value, the cylinder will turn clockwise microscopically to stress the welds at both contact points and make both forces of friction increase to their maximum values. A comparison: To make a four-legged table start to slide across the fl oor, you must push on it hard enough to counterbalance the maximum static friction forces on all four legs together.

When it is on the verge of slipping, the cylinder is in equilibrium.

Fx∑ = 0: f n ns1 2 1= = μ and f ns2 2= μ

Fy∑ = 0: P n f Fg+ + =1 2

τ∑ = 0: P f f= +1 2

As P grows so do f1 and f

2 .

Therefore, since μs = 1

2, f

n1

1

2= and f

n n2

2 1

2 4= =

then P nn

Fg+ + =11

4 (1) and P

n nn= + =1 1

12 4

3

4 (2)

So P n Fg+ =5

4 1 becomes P P Fg+ ⎛⎝

⎞⎠ =5

4

4

3 or

8

3P Fg=

Therefore,

P Fg= 3

8

P12.53 (a) F k L= ( )Δ , Young’s modulus is Y = = ( )F A

L L

FL

A Li

i/

/Δ Δ

Thus,

Y = kL

Ai and k

A

Li

= Y

(b) W Fdx kx dxA

Lxdx A

LL L

i

L

= − = − −( ) = = ( )∫ ∫ ∫0 0 0

2Δ Δ Δ ΔYY

22Li

FIG. P12.52



P1

2

3

3.33 NP

P

FIG. P12.54(a)

P12.54 (a) Take both balls together. Their weight is 3.33 N and their CG is at their contact point.

Fx∑ = 0: + − =P P3 1 0

Fy∑ = 0: + − =P2 3 33 0. N P2 3 33= . N

τ A∑ = 0: − + −P R P R3 2 3 33. NR R+( ) +cos .45 0°

P R R+ +( ) =cos .2 45 0 01 °

Substituting,

− + ( ) − ( ) +( )+ +

P R R R

P R1

1

3 33 3 33 1 45 0

1 2

. . cos .N N °

ccos .

. cos . cos .

45 0 0

3 33 45 0 2 45 01

1

°

° °

( ) =( ) =N P

P == =1 67 1 673. .N so NP

(b) Take the upper ball. The lines of action of its weight, of P1,

and of the normal force n exerted by the lower ball all go through its center, so for rotational equilibrium there can be no frictional force.

Fx∑ = 0: n Pcos .45 0 01° − =

n = =1 672 36

.

cos.

N

45.0N

°

Fy∑ = 0: n sin . .45 0 1 67 0° − =N gives the same result.

P12.55 Fy∑ = 0: + − + =380 320 0N NFg

Fg = 700 N

Take torques about her feet:

τ∑ = 0: − ( ) + ( )380 2 00 700N m N. x

+( ) =320 0 0N

x = 1 09. m

P12.56 The tension in this cable is not uniform, so this becomes a fairly diffi cult problem.

dL

L

F

A=

Y At any point in the cable, F is the weight of cable below that point. Thus, F gy= µ where µ is the

mass per unit length of the cable.

Then,

∆ydL

Ldy

g

YAydy

gL

YA

L L

ii i

= ⎛⎝

⎞⎠ = =∫ ∫

0 0

21

2

µ µ

∆y = ( )( )( )×( ) × −

1

2

2 40 9 80 500

2 00 10 3 00 10

2

11 4

. .

. .(( ) = =0 049 0 4 90. .m cm

1.67 N

n cos 45.0°

n sin 45.0°

P1

FIG. P12.54(b)

FIG. P12.55


334 Chapter 12

P12.57 (a) F mt

= ⎛⎝

⎞⎠ = ( ) −( )

=ΔΔ

v1 00

10 0 1 00

0 002.

. .

.kg

m s

s44 500 N

(b) stress = =( )( ) = ×F

A

4 500

0 1004 50 106N

0.010 m mN m2

..

(c) Yes. This is more than suffi cient to break the board.

P12.58 Let θ represent the angle of the wire with the vertical. The radius of the circle of motion is r = ( )0 850. sinm θ .

For the mass:

F ma mr

mr

T m

r r∑ = = =

= ( )[ ]

v22

20 850

ω

θ θ ωsin . sinm

Further, T

A= ⋅( )Y strain or T A= ⋅( )Y strain

Thus, A mY ⋅( ) = ( )strain m0 850 2. ω , giving

ωπ

= ⋅( )( ) =

×( )−A

m

Y strain

m

m

0 850

3 90 10 7 004 2

.

. . ××( ) ×( )( )( )

−10 1 00 10

1 20 0 850

10 3N m

kg m

2 .

. .

or ω = 5 73. rad s

P12.59 (a) If the acceleration is a, we have Px = ma

and P n Fy g+ − = 0. Taking the origin at the center of gravity, the torque equation gives

P L d P h ndy x−( ) + − = 0

Solving these equations, we fi nd

PF

Ld

ah

gyg= −

⎛⎝⎜

⎞⎠⎟

(b) If Py = 0, then d

ah

g= =

( )( )=

2 00 1 50

9 800 306

. .

..

m s m

m sm

2

2 .

(c) Using the given data, Px = −306 N and Py = 553 N.

Thus,�P i j= − +( )306 553ˆ ˆ N .

T

rmg

FIG. P12.58

hhP

CGCGCG

ddHL

Fyn Fgn

FIG. P12.59



P12.60 When the car is on the point of rolling over, the normal force on its inside wheels is zero.

F may y∑ = : n mg− = 0

F max x∑ = : fm

R= v2

Take torque about the center of mass: f h nd− =2

0.

Then by substitution m

Rh

mgdvmax2

20− = vmax = gdR

h2

A wider wheelbase (larger d) and a lower center of mass (smaller h) will reduce the risk of rollover.

ANSWERS TO EVEN PROBLEMS

P12.2 F R Fy y g+ − = 0; F Rx x− = 0; F F Fy g x��

�cos cos sinθ θ θ− ⎛⎝⎜

⎞⎠⎟ − =

20

P12.4 see the solution

P12.6 0.750 m

P12.8 (a) 9.00 g (b) 52.5 g (c) 49.0 g

P12.10 (a) 1.04 kN at 60° upward and to the right. (b) �F i jsurface N N= ( ) + ( )370 900ˆ ˆ

P12.12 (a) fm g m gx

L= +⎡

⎣⎢⎤⎦⎥

1 2

2cotθ ; n m m gg = +( )1 2 (b) μ

θ=

+( )+

m m d L

m m1 2

1 2

2/ / cot

P12.14 (a) 392 N (b) 339 N to the right (c) 0 (d) 0 (e) 392 N (f ) 339 N to the right (g) Both are equally accurate and essentially equally simple. We appear to have four equations, but only three are independent. We can determine three unknowns.

P12.16 (a) 35.5 kN (b) 11.5 kN to the right (c) 4.19 kN down

P12.18 (a) 859 N (b) 104 kN at 36.9° above the horizontal to the left

P12.20 (a) see the solution (b) θ decreases (c) R decreases

P12.22 1.0 × 1011 N�m2

P12.24 (a) 73.6 kN (b) 2.50 mm

P12.26 ~ 1 cm

P12.28 9 85 10 5. × −

P12.30 0 029 3. mm

P12.32 (a) −0.053 8 m3 (b) 1.09 × 103 kg�m3 (c) With only a 5% change in volume in this extreme case, liquid water is indeed nearly incompressible in biological and student-laboratory situations.

mg

h

mgf

d

FIG. P12.60


336 Chapter 12

P12.34 The weight must be 508 N or more. The person could be a child. We assume the stove is a uni-form box with feet at its corners. We ignore the masses of the backsplash and the oven door. If the oven door is heavy, the minimum weight for the person might be somewhat less than 508 N.

P12.36 (a) 0.400 mm (b) 40.0 kN (c) 2.00 mm (d) 2.40 mm (e) 48.0 kN

P12.38 (a) See the solution. The weight of the uniform gate is 392 N. It is 3.00 m wide. The hinges are separated vertically by 1.80 m. The bucket of grain weighs 50.0 N. One of the hinges, which we suppose is the upper one, supports the whole weight of the gate. Find the components of the forces that both hinges exert on the gate. (b) The upper hinge exerts A = 410 N to the left andB = 442 N up. The lower hinge exerts C = 410 N to the right.

P12.40 1 46. kN; 1 33 2 58. ˆ . ˆi j+( ) kN

P12.42 (a) 2.71 kN (b) 2.65 kN (c) You should lift “with your knees” rather than “with your back.” In this situation, you can make the compressional force in your spine about ten times smaller by bending your knees and lifting with your back as straight as possible.

P12.44 (a) see the solution (b) 60.0°

P12.46 (a) 120 N (b) 0.300 (c) 103 N at 31.0° above the horizontal to the right

P12.48 Assume a strut exerts on a pin a force with a component perpendicular to the length of the strut. Then the pin must exert a force on the strut with a perpendicular component of this size. For translational equilibrium, the pin at the other end of the strut must also exert the same size force on the strut in the opposite direction. Then the strut will feel two torques about its center in the same sense. It will not be in equilibrium, but will start to rotate. The contradiction proves that we were wrong to assume the existence of the perpendicular force. The strut can exert on the pin only a force parallel to its length.

P12.50 (a) 66.7 N (b) increasing at 0.125 N�s

P12.52 If either static friction force were at less than its maximum value, the cylinder would rotate by a microscopic amount to put more stress on some welds and to bring that friction force to its maxi-mum value. P = 3F

g �8

P12.54 (a) P1 = 1.67 N; P2 3 33= . N; P3 1 67= . N (b) 2.36 N

P12.56 4.90 cm

P12.58 5.73 rad�s

P12.60 See the solution. A wider wheelbase (larger d) and a lower center of mass (smaller h) will reduce the risk of rollover.


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