Mechanical Engineering Chapter 9 Problem The Morse Taper Used Support Taper Spindle The Workpiece Machinists Lathe

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Problem 9.122* The “Morse taper” is used to support

the workpiece on a machinist’s lathe. The taper is driven

into the spindle and is held in place by friction. If the

spindle exerts a uniform pressure pD15 psi on the taper

and sD0.2, what couple must be exerted about the

axis of the taper to loosen it?

Taper

2 in

Spindle

1.25 in

9 in

Solution: The outer radius of the taper is roD1 in, and the inner

radius is riD0.6.5 in. The angle of the taper is

check. The normal force on the taper is NDpA D689.8 lb, and the

axial force on the taper is FDNcos ˛D28.72 lb. The taper is equiv-

Problem 9.123 In Active Example 9.9, suppose that

the left ﬁxed cylinder is replaced by a pully. Assume

that the tensions in the rope on each side of the pulley

are approximately equal. What is the smallest force the

woman needs to exert on the rope to support the

stationary box?

Problem 9.124 Suppose that you want to lift a 50-lb

crate off the ground by using a rope looped over a tree

limb as shown. The coefﬁcient of static friction between

the rope and the limb is 0.2, and the rope is wound 135°

around the limb. What force must you exert to begin

lifting the crate?

Problem 9.125 Winches are used on sailboats to help

support the forces exerted by the sails on the ropes

(sheets) holding them in position. The winch shown is a

post that will rotate in the clockwise direction (seen from

above), but will not rotate in the counterclockwise direc-

tion. The sail exerts a tension TSD800 N on the sheet,

which is wrapped two complete turns around the winch.

The coefﬁcient of static friction between the sheet and

the winch is sD0.2. What tension TCmust the crew

member exert on the sheet to prevent it from slipping

on the winch?

TsD800 N sD0.2

ˇD4

Solving,

TcD64.8N

Problem 9.126 The coefﬁcient of kinetic friction

between the sheet and the winch in Problem 9.125 is

kD0.16. If the crew member wants to let the sheet slip

at a constant rate, releasing the sail, what initial tension

TCmust he exert on the sheet as it begins slipping?

766

Problem 9.127 The box Aweights 20 lb. The rope

is wrapped one and one-fourth turns around the ﬁxed

wooden post. The coefﬁcients of friction between the

rope and post are sD0.15 and kD0.12.

(a) What minimum force does the man need to exert

to support the stationary box?

(b) What force would the man have to exert to raise

the box at a constant rate?

Solution:

Problem 9.128 The weight of the block Ais W. The

disk is supported by a smooth bearing. The coefﬁcient

of kinetic friction between the disk and the belt is k.

What couple Mis necessary to turn the disk at a constant

rate?

Problem 9.129 The couple required to turn the wheel

of the exercise bicycle is adjusted by changing the

weight W. The coefﬁcient of kinetic friction between

the wheel and the belt is k. Assume the wheel turns

clockwise.

(a) Show that the couple Mrequired to turn the wheel

is MDWR⊲1e3.4k⊳.

(b) If WD40 lb and kD0.2, what force will the

scale S indicate when the bicycle is in use?

15°

Problem 9.130 The box Bweighs 50 lb. The coefﬁ–

cient of friction between the cable and the ﬁxed round

supports are sD0.4 and kD0.3.

(a) What is the minimum force Frequired to support

the box?

(b) What force Fis required to move the box upward

at a constant rate? BF

Problem 9.131 The coefﬁcient of static friction

between the 50-lb box and the inclined surface is 0.10.

The coefﬁcient of static friction between the rope and the

ﬁxed cylinder is 0.05. Determine the force the woman

must exert on the rope to cause the box to start moving

up the inclined surface.

20⬚30⬚

45⬚

768

Problem 9.132 In Problem 9.131, what is the mini-

mum force the woman must exert on the rope to hold

the box in equilibrium on the inclined surface?

Solution: See 9.131 – Change the friction force

F%:T2cos 25°Cf50 lb sin 20°D0

F–:T2sin 25°CN50 lb cos 20°D0

fD0.1N

T2DTe0.05⊲7/12⊳

Solving TD13.10 lb

50 lb

Problem 9.133 Blocks Band Ceach have a mass of

20 kg. The coefﬁcient of static friction at the contacting

surfaces is 0.2. Block Ais suspended by a rope that

passes over a ﬁxed cylinder and is attached to block B.

The coefﬁcient of static friction between the rope and

the cylinder is 0.3. What is the largest mass block Acan

have without causing block Bto slip the left?

20⬚

Problem 9.134 If the force Fin Example 9.10 is

increased to 400 N, what are the largest values of the

couples MAand MBfor which the belt will not slip?

Solution: From Example 9.10, bD500 mm, sD0.8, RaD

200 mm, RbD100 mm. The angle of contact for pulley Ais ˇaD

C2˛. The angle of contact for pulley Bis ˇbD2˛, where

˛Dsin1RaRb

bDsin10.1

0.5D0.2014 radians.

The belt contact is less for pulley B, so it is most likely to slip ﬁrst.

The couples are in opposition so that the tension in the upper belt is

greater than the tension in the lower belt: For belt B:

Tupper DTloweresˇbD8.945Tlower.

770

Problem 9.135 The spring exerts a 320-N force on the

left pulley. The coefﬁcient of static friction between the

ﬂat belt and the pulleys is sD0.5. The right pulley

cannot rotate. What is the largest couple Mthat can be

exerted on the left pulley without causing the belt to

slip?

100 mm

260 mm

40 mm

Solution: The angle of the belt relative to the horizontal is

Since the angle of contact is less on the right pulley, it should slip

Problem 9.136 The weight of the box is WD30 lb,

and the force Fis perpendicular to the inclined surface.

The coefﬁcient of static friction between the box and the

inclined surface is sD0.2.

(a) If FD30 lb, what is the magnitude of the friction

force exerted on the stationary box?

(b) If FD10 lb, show that the box cannot remain at

rest on the inclined surface.

20°

Problem 9.137 In Problem 9.136, what is the smallest

force Fnecessary to hold the box stationary on the

inclined surface?

Problem 9.138 Blocks Aand Bare connected by a

horizontal bar. The coefﬁcient of static friction between

the inclined surface and the 400-lb block Ais 0.3. The

coefﬁcient of static friction between the surface and the

300-lb block Bis 0.5. What is the smallest force Fthat

will prevent the blocks from slipping down the surface?

45°

Solution: The (horizontal) connecting bar exerts a component of

772

Problem 9.139 What force Fis necessary to cause the

blocks in Problem 9.138 to start sliding up the plane?

flink DB

⊲1sB⊳⊲sin ˛CsBcos ˛⊳ D636.4lb.

Isolate A. The resultant force on Ais

Fflink DA⊲sin ˛CsAcos ˛⊳ sAflink D176.78 lb.

The resultant force required to cause the blocks to start to move up

the plane is FD636.4C176.8D813.2lb

SANA

Problem 9.140 The masses of crates Aand Bare 25 kg

and 30 kg, respectively. The coefﬁcient of static friction

between the contacting surfaces is sD0.34. What is

the largest value of ˛for which the crates will remain

in equilibrium? A

Problem 9.141 The side of a soil embankment has a

45°slope (Fig. a). If the coefﬁcient of static friction of

soil on soil is sD0.6, will the embankment be stable

or will it collapse? If it will collapse, what is the smallest

slope that can be stable?

Strategy: Draw a free-body diagram by isolating part

of the embankment as shown in Fig. b.

(a)

45°

(b)

774

Problem 9.142 The mass of the van is 2250 kg, and

the coefﬁcient of static friction between its tires and the

road is 0.6. If its front wheels are locked and its rear

wheels can turn freely, what is the largest value of ˛for

which it can remain in equilibrium?

1 m

Solution: Choose a coordinate system with the xaxis parallel

to the incline. The weight of the van is WDmg D22072.5 N. The

moment about the point of contact of the rear wheels is

MRD⊲31.2⊳W cos ˛C1Wsin ˛3ND0,

The sum of the forces parallel to the inclined surface is

s1.8

3cos ˛Cs

31sin ˛D0,

from which

˛Dtan11.8s

3sDtan1⊲0.45⊳D24.2°

1 m

1.2 m

1.8 m

Problem 9.143 In Problem 9.142, what is the largest

value of ˛for which the van can remain in equilibrium

if it points up the slope?

Problem 9.144 The shelf is designed so that it can be

placed at any height on the vertical beam. The shelf is

supported by friction between the two horizontal cylin-

ders and the vertical beam. The combined weight of the

shelf and camera is W. If the coefﬁcient of static friction

between the vertical beam and the horizontal cylinders

is s, what is the minimum distance bnecessary for the

shelf to stay in place?

776