Mechanical Engineering Chapter 9 Problem Active Example Suppose That The Coefcient Static Friction Between The Crateand

Problem 9.1 In Active Example 9.1, suppose that the

coefﬁcient of static friction between the 180-lb crate

and the ramp is sD0.3. What is the magnitude of

the smallest horizontal force the rope must exert on the

crate to prevent it from sliding down the ramp?

Problem 9.2 A person places a 2-lb book on a table

that is tilted at 5°relative to the horizontal. She ﬁnds that

if she exerts a very small force on the book as shown,

the book remains in equilibrium, but if she removes the

force, the book slides down the table. What force would

she need to exert on the book (in the direction parallel

to the table) to cause it to slide up the table?

15⬚

Problem 9.3 A student pushes a 200-lb box of books

across the ﬂoor. The coefﬁcient of kinetic friction

between the carpet and the box is kD0.15.

(a) If he exerts the force Fat angle ˛D25°, what is

the magnitude of the force he must exert to slide

the box across the ﬂoor?

(b) If he bends his knees more and exerts the force

Fat angle ˛D10°, what is the magnitude of the

force he must exert to slide the box?

a

F

N

Problem 9.4 The 2975-lb car is parked on a sloped

street. The brakes are applied to both its front and rear

wheels.

(a) If the coefﬁcient of static friction between the car’s

tires and the road is sD0.8, what is the steepest

slope (in degrees relative to the horizontal) on

which the car could remain in equilibrium?

(b) If the street were icy and the coefﬁcient of static

friction between the car’s tires and the road was

sD0.2, what is the steepest slope on which the

car could remain in equilibrium?

686

Problem 9.5 The truck’s winch exerts a horizontal

force on the 200-kg crate in an effort to pull it down

the ramp. The coefﬁcient of static friction between the

crate and the ramp is sD0.6.

(a) If the winch exerts a 200-N horizontal force on the

crate, what is the magnitude of the friction force

exerted on the crate by the ramp?

(b) What is the magnitude of the horizontal force the

winch must exert on the crate to cause it to start

moving down the ramp?

20⬚

Problem 9.6 The device shown is designed to position

pieces of luggage on a ramp. It exerts a force parallel

to the ramp. The suitcase weighs 40 lb. The coefﬁcients

of friction between the suitcase and ramp are sD0.20

and kD0.18.

(a) Will the suitcase remain stationary on the ramp

when the device exerts no force on it?

(b) What force must the device exert to push the

suitcase up the ramp at a constant speed?

20⬚

688

Problem 9.7 The coefﬁcient of static friction between

the 50-kg crate and the ramp is sD0.35. The

unstretched length of the spring is 800 mm, and the

spring constant is kD660 N/m.

What is the minimum value of xat which the crate can

remain stationary on the ramp?

k

x

FsD⊲660 N/m⊳⊲0.8mx⊳

Fs

490.5 N

Problem 9.8 The coefﬁcient of kinetic friction bet-

ween the 40-kg crate and the slanting ﬂoor is kD0.3.

If the angle ˛D20°, what tension must the person exert

on the rope to move the crate at constant speed?

10⬚

a

Solution:

fD0.3N

T

392.4 N

f

Problem 9.9 In Problem 9.8, for what angle ˛is the

tension necessary to move the crate at constant speed a

minimum? What is the necessary tension?

fD0.3N

)TD184.1N

)

TD176.3N

Problem 9.10 Box Aweighs 100 lb, and box Bweighs

30 lb. The coefﬁcients of friction between box Aand the

ramp are sD0.30 and kD0.28. What is the magni-

tude of the friction force exerted on box Aby the ramp?

A

B

690

Problem 9.11 In Problem 9.10, box Aweighs 100 lb,

and the coefﬁcients of friction between box Aand the

ramp are sD0.30 and kD0.28. For what range

of the weights of the box Bwill the system remain

stationary?

D100⊲sin 30°C0.3 cos 30°⊳D75.98 lb

For impending motion down the ramp:

FDAsin ˛BMIN sAcos ˛D0,

from which

BDA⊲sin ˛scos ˛⊳

D100⊲sin 30°0.3 cos 30°⊳D24.02 lb

N

Problem 9.12 The mass of the box on the left is 30 kg,

and the mass of the box on the right is 40 kg. The coefﬁ–

cient of static friction between each box and the inclined

surface is sD0.2. Determine the minimum angle ˛for

with the boxes will remain stationary.

30⬚

a

Solution: If the boxes slip when ˛is decreased, they will slip

toward the right. Assume that slip toward the right impends, the free

body diagrams are as shown.

FxDT0.2N

A⊲30⊳⊲9.81⊳sin ˛D0,(1)

0.2 NA

0.2 NB

NANB

α

x

y

yTT

30°

Problem 9.13 The coefﬁcient of kinetic friction between

the 100-kg box and the inclined surface is 0.35. Deter-

mine the tension Tnecessary to pull the box up the

T

692

Problem 9.14 The box is stationary on the inclined

surface. The coefﬁcient of static friction between the

box and the surface is s.

(a) If the mass of the box is 10 kg, ˛D20°,ˇD30°,

and sD0.24, what force Tis necessary to start

the box sliding up the surface?

T

β

gD9.81 m/s2

Fx:Tcos ˇCfCmg sin ˛D0

Problem 9.15 To explain observations of ship launch-

ings at the port of Rochefort in 1779, Coulomb analyzed

the system shown in Problem 9.14 to determine the

minimum force Tnecessary to hold the box stationary

on the inclined surface. Show that the result is

694

Problem 9.16 Two sheets of plywood Aand Blie on

the bed of a truck. They have the same weight W, and

the coefﬁcient of static friction between the two sheets

of wood and between sheet Band the truck bed is s.

(a) If you apply a horizontal force to sheet Aand apply

no force to sheet B, can you slide sheet Aoff the

truck without causing sheet Bto move? What force

is necessary to cause sheet Ato start moving?

(b) If you prevent sheet Afrom moving by applying

a horizontal force on it, what horizontal force on

sheet Bis necessary to start it moving?

B

A

FDf

AB DsW.

(b) The force on Bis the friction between Aand Band the friction

between Band the truck bed. Thus the force required to start B

in motion is

FBDf

AB Cf

BT D3sW.

fBT

FB

2W

Problem 9.17 The weights of the two boxes are W1D

100 lb and W2D50 lb. The coefﬁcients of kinetic fric-

tion between the left box and the inclined surface are

sD0.12 and kD0.10. Determine the tension the

man must exert on the rope to pull the boxes upward

at a constant rate.

30⬚

W2

W1

Problem 9.18 In Problem 9.17, for what range of

tensions exerted on the rope by the man will the boxes

remain stationary?

fD0.12 N

Next solve for the smallest force Tmin. We need to turn the friction

89.6lb<T<110.4lb

Problem 9.19 Each box weighs 10 lb. The coefﬁcient

of static friction between box Aand box Bis 0.24, and

the coefﬁcient of static friction between box Band the

inclined surface is 0.3. What is the largest angle ˛for

which box Bwill not slip?

A

B

696

Problem 9.20 The masses of the boxes are mAD

15 kg and mBD60 kg. The coefﬁcient of static friction

between boxes Aand Band between box Band the

inclined surface is 0.12. What is the largest force Ffor

which the boxes will not slip? FA

B

f1D0.12N1,f

2D0.12N2

Solving we ﬁnd FD267 N

20°

N1

f1

B

588.6 N

Problem 9.21 In Problem 9.20, what is the smallest

force Ffor which the boxes will not slip?

Problem 9.22 In Example 9.2, what clockwise couple

Mwould need to be applied to the disk to cause it to

rotate at a constant rate in the clockwise direction?

C

M

weighs 20 lb. The homogeneous disk weighs 30 lb. The

coefﬁcient of kinetic friction between the disk and the

sloping surface is kD0.24. What is the magnitude of

the couple that would need to be applied to the disk

to cause it to rotate at a constant rate in the clockwise

direction?

AB

1 ft

MD11.2 ft-lb.

698

c

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Problem 9.24 The homogeneous horizontal bar AB

weighs 20 lb. The homogeneous disk weighs 30 lb. The

coefﬁcient of kinetic friction between the disk and the

sloping surface is kD0.24. What is the magnitude of

the couple that would need to be applied to the disk to

cause it to rotate at a constant rate in the counterclock-

wise direction?

AB

5 ft

1 ft

Problem 9.25 The mass of the bar is 4 kg. The coef-

ﬁcient of static friction between the bar and the ﬂoor is

0.3. Neglect friction between the bar and the wall.

(a) If ˛D20°, what is the magnitude of the friction

force exerted on the bar by the ﬂoor?

(b) What is the maximum angle ˛for which the bar

will not slip?

1 m

a

Solving: fBD7.14 N,f

max D11.77 N

Since fB<f

max,f

BD7.14 N

1.0 m

Problem 9.26 In Problem 9.25, suppose that the coef-

ﬁcient of static friction between the bar and the ﬂoor

and between the 4-kg bar and the wall is 0.3. What is

the maximum angle ˛for which the bar will not slip?

fBD0.3NB,f

AD0.3NB

Fy:NBCfA39.24 N D0

MB:⊲39.24 N⊳⊲0.5m⊳sin ˛NA⊲1.0m⊳cos ˛

fA⊲1.0m⊳sin ˛D0

NA

1.0 m

700

Problem 9.27 The ladder and the person weigh 30 lb

and 180 lb, respectively. The center of mass of the 12-ft

ladder is at its midpoint. The angle ˛D30°. Assume that

the wall exerts a negligible friction force on the ladder.

(a) If xD4 ft, what is the magnitude of the friction

force exerted on the ladder by the ﬂoor?

(b) What minimum coefﬁcient of static friction

between the ladder and the ﬂoor is necessary for

the person to be able to climb to the top of the

ladder without slipping?

x

a

Problem 9.28 In Problem 9.27, the ladder and the per-

son weigh 30 lb and 180 lb, respectively. The center of

mass of the 12-ft ladder is at its midpoint. The coefﬁcient

of static friction between the ladder and the ﬂoor is sD

0.5. What is the largest value of the angle ˛for which

the person could climb to the top of the ladder without

it slipping?

Problem 9.29 In Problem 9.27, the ladder and the per-

son weigh 30 lb and 180 lb, respectively. The center of

mass of the 12-ft ladder is at its midpoint. The coefﬁcient

of static friction between the ladder and the ﬂoor is 0.5

and the coefﬁcient of friction between the ladder and the

wall is 0.3. What is the largest value of the angle ˛for

which the person could climb to the top of the ladder

without it slipping? Compare your answer to the answer

to Problem 9.28.

Solution:

xD⊲12 ft⊳sin ˛

Fx:fBNAD0

Fy:NB210 lb CfAD0

MB:NA⊲12 ft cos ˛⊳ 30 lb ⊲6 ft sin ˛⊳

180 lb xCfAxD0

fBD0.5NB,f

AD0.3NA

fA

NA

180 lb

α

702

Problem 9.30 The disk weighs 50 lb and the bar

weighs 25 lb. The coefﬁcients of friction between the

disk and the inclined surface are sD0.6 and kD0.5.

(a) What is the largest couple Mthat can be applied

to the stationary disk without causing it to

start rotating?

(b) What couple Mis necessary to rotate the disk at a

constant rate?

M

5 in

20 in

fD0.6N

)MD162.4lbin

(b) Steady rotation. Replace the last equation with

fD0.5N )MD135.3lbin

30°

M

f

N

B2

Ax

25 lb

Problem 9.31 The radius of the 40-kg homogeneous

cylinder is R = 0.15 m. The slanted wall is smooth

and the angle ˛D30°. The coefﬁcient of static friction

between the cylinder and the ﬂoor is sD0.2. What is

the largest couple Mthat can be applied to the cylinder

without causing it to slip?

M

R

a

Problem 9.32 The homogeneous cylinder has weight

W. The coefﬁcient of static friction between the cylinder

and both surfaces is s. What is the largest couple M

that can be applied to the cylinder without causing it to

slip? (Assume that the cylinder slips before rolling up

M

R

Problem 9.33 The homogeneous cylinder has weight

W. The coefﬁcient of static friction between the cylinder

and both surfaces is s. What is the minimum value of

sfor which the couple Mwill cause the cylinder to

roll up the inclined surface without slipping?

M

R

a

704