Mechanical Engineering Chapter 19 Problem The Moment Inertia The Rotor The Medical Centrifuge Kgm The Rotor

Problem 19.1 The moment of inertia of the rotor of

0.8 N-m on it.

(a) How much work has the motor done on the rotor

when the rotor has rotated through four revolu-

tions?

(b) What is the rotor’s angular velocity (in rpm) when

it has rotated through four revolutions?

Problem 19.2 The 4-lb slender bar is 2 ft in length. It

started from rest in an initial position relative to the iner-

tial reference frame. When it is in the position shown,

the velocity of the end Ais 2i+6j(ft/s) and the bar has

a counterclockwise angular velocity of 12 rad/s. How

much work was done on the bar as it moved from its

initial position to its present position?

y

B

560

Problem 19.3 The 20-kg disk is at rest when the

it has rotated through four revolutions (a) by applying

the equation of angular motion M =1α, and (b) by

applying the principle of work and energy.

0.25 m

Solution:

Problem 19.4 The space station is initially not rotat-

ing. Its reaction control system exerts a constant couple

on it until it has rotated 90◦, then exerts a constant couple

of the same magnitude in the opposite direction so that

its angular velocity has decreased to zero when it has

undergone a total rotation of 180◦. The maneuver takes

6 hours. The station’s moment of inertia about the axis

of rotation is I=1.5×1010 kg-m2. How much work is

done in performing this maneuver? In other words, how

much energy had to be expended in the form of reaction

control fuel?

562

Problem 19.5 The helicopter’s rotor starts from rest.

Suppose that its engine exerts a constant 1200 ft-lb cou-

ple on the rotor and aerodynamic drag is negligible. The

rotor’s moment of inertia is I=400 slug-ft2.

(a) Use work and energy to determine the magnitude

of the rotor’s angular velocity when it has rotated

through ﬁve revolutions.

(b) What average power is transferred to the rotor

while it rotates through ﬁve revolutions?

Problem 19.6 The helicopter’s rotor starts from rest.

The moment exerted on it (in N-m) is given as a function

of the angle through which it has turned in radians by

M=6500 −20θ. The rotor’s moment of inertia is I=

540 kg-m2. Determine the rotor’s angular velocity (in

rpm) when it has turned through 10 revolutions.

Solution: We will integrate to ﬁnd the work.

Problem 19.7 During extravehicular activity, an astro-

naut’s angular velocity is initially zero. She activates two

is 45 kg-m2. Use the principle of work and energy to

determine the angle through which she has rotated when

her angular velocity reaches 15◦per second.

T

Problem 19.8 The 8-kg slender bar is released from

rest in the horizontal position 1 and falls to position 2.

(a) How much work is done by the bar’s weight as it

falls from position 1 to position 2?

(b) How much work is done by the force exerted on

the bar by the pin support as the bar falls from

position 1 to position 2?

A

2 m

y

x

1

564

Problem 19.9 The 20-lb bar is released from rest in the

2

M

y

Solution: We will use the energy equation in the form

T1+V1+U12 =T2+V2

Problem 19.10 The object consists of an 8-lb slender

2

45⬚

y

Problem 19.11 The object consists of an 8-lb slender

bar welded to a 12-lb circular disk. The object is released

from rest in position 1. Determine the xand ycompo-

nents of force exerted on the object by the pin support

when it is in position 2.

5 in

22 in

1

2

45⬚

x

A

y

Solution: We ﬁrst determine the moment of inertia about the ﬁxed

point Aand the distance from Ato the center of mass.

22

We now write the equations of motion and the work energy equation

Problem 19.12 The mass of each box is 4 kg. The

radius of the pulley is 120 mm and its moment of inertia

is 0.032 kg-m2. The surfaces are smooth. If the system is

released from rest, how fast are the boxes moving when

the left box has moved 0.5 m to the right?

566

Problem 19.13 The mass of each box is 4 kg. The

radius of the pulley is 120 mm and its moment of iner-

tia is 0.032 kg-m2. The coefﬁcient of kinetic friction

between the boxes and the surfaces is µk=0.12. If the

system is released from rest, how fast are the boxes mov-

ing when the left box has moved 0.5 m to the right? 30⬚

Problem 19.14 The 4-kg bar is released from rest in

the horizontal position 1 and falls to position 2. The

unstretched length of the spring is 0.4 m and the spring

constant is k=20 N/m. What is the magnitude of the

bar’s angular velocity when it is in position 2. 60⬚

1

A

0.6 m 1 m

Problem 19.15 The moments of inertia of gears

that can turn freely on their pin supports are IA=

0.002 kg-m2and IB=0.006 kg-m2. The gears are at

rest when a constant couple M=2 N-m is applied to

gear B. Neglecting friction, use principle of work and

60 mm

A

M

0

The kinetic energy is T2=1

2IAω2

A+1

2IBω2

B,

where ωB=−rA

rBωA,from which

0.09 ωA=399.5 rad/s

Problem 19.16 The moments of inertia of gears A

and Bare IA=0.02 kg-m2and IB=0.09 kg-m2.

Gear Ais connected to a torsional spring with

constant k=12 N-m/rad. If gear Bis given an initial

counterclockwise angular velocity of 10 rad/s with the

torsional spring unstretched, through what maximum

counterclockwise angle does gear Brotate? 200 mm

AB

=0.731 rad =41.9◦.

568

c

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Problem 19.17 The moments of inertia of three

100 mm

From the principle of work and energy: U=T2−T1, where T1=0

since the pulleys start from a stationary position. The work done is

U=θA

0

Mdθ =2(2π)(10)=40πN-m.

0.0065 =139.0 rad/s

Problem 19.18 Model the arm ABC as a single rigid

body. Its mass is 300 kg, and the moment of inertia

from rest with its center of mass 2 m above the ground

1.4 rad/s. How much work do the hydraulic cylinders do

on the arm in moving it from position 1 to position 2?

1.40 m

0.70 m

2.25 m

A

C

Solution: From the principle of work and energy: U=T2−T1,

where T1=0 since the system starts from rest. The work done is

U=Ucylinders −mg(h2−h1), and the kinetic energy is

Problem 19.19 The mass of the circular disk is 5 kg

and its radius is R=0.2 m. The disk is stationary when

a constant clockwise couple M=10 N-m is applied to

it, causing the disk to roll toward the right. Consider the

disk when its center has moved a distance b=0.4m.

(a) How much work has the couple Mdone on the

disk?

(b) How much work has been done by the friction force

exerted on the disk by the surface?

(c) What is the magnitude of the velocity of the center

of the disk?

b

M

Problem 19.20 The mass of the homogeneous cylin-

drical disk is m=5 kg and its radius is R=0.2 m. The

angle β=15◦. The disk is stationary when a constant

clockwise couple M=10 N-m is applied to it. What is

the velocity of the center of the disk when it has moved

a distance b=0.4 m? (See Active Example 19.1.)

M

R

b

β

570

Problem 19.21 The mass of the stepped disk is 18 kg

and its moment of inertia is 0.28 kg-m2. If the disk is

released from rest, what is its angular velocity when the

center of the disk has fallen 1 m?

0.2

m

0.1

m

176.6=1

2mv2+1

2Iω2

Problem 19.22 The 100-kg homogenous cylindrical

disk is at rest when the force F=500 N is applied to a

cord wrapped around it, causing the disk to roll. Use the

principle of work and energy to determine the angular

velocity of the disk when it has turned one revolution.

F

300 mm

Solution: From the principle of work and energy: U=T2−T1,

Problem 19.23 The 1-slug homogenous cylindrical

disk is given a clockwise angular velocity of 2 rad/s with

the spring unstretched. The spring constant is k=3 lb/ft.

If the disk rolls, how far will its center move to the right? 1 ft

k

U=−1

2kS2.

The kinetic energy is

T1=1

2Iω2+1

2mv2.

By inspection v=ωR, from which

T1=1

2m

2R2+mR2ω2=3

4mR2ω2=0.75(22)=3 ft-lb,

U=−T1,−3

2S2=−3,

from which S=√2=1.414 ft.

Problem 19.24 The system is released from rest. The

moment of inertia of the pulley is 0.03 slug-ft2. The

slanted surface is smooth. Determine the magnitude of

the velocity of the 10-lb weight when it has fallen 2 ft.

10 lb

6 in

20⬚

5 lb

Solution: Use conservation of energy

572

Problem 19.25 The system is released from rest. The

moment of inertia of the pulley is 0.03 slug-ft2. The

coefﬁcient of kinetic friction between the 5-lb weight

and the slanted surface is µk=0.3. Determine the mag-

nitude of the velocity of the 10-lb weight when it has

fallen 2 ft.

10 lb

6 in

20⬚

5 lb

Solution: Use work energy.

Problem 19.26 Each of the cart’s four wheels weighs

3 lb, has a radius of 5 in, and has moment of inertia

I=0.01 slug-ft2. The cart (not including its wheels)

weighs 20 lb. The cart is stationary when the constant

horizontal force F=10 lb is applied. How fast is the

cart going when it has moved 2 ft to the right?

F

Problem 19.27 The total moment of inertia of car’s

two rear wheels and axle is IR, and the total moment of

inertia of the two front wheels is IF. The radius of the

tires is R, and the total mass of the car, including the

wheels, is m. The car is moving at velocity v0when

the driver applies the brakes. If the car’s brakes exert a

constant retarding couple Mon each wheel and the tires

do not slip, determine the car’s velocity as a function

of the distance sfrom the point where the brakes are

applied.

s

Solution: When the car rolls a distance s, the wheels roll through

Problem 19.28 The total moment of inertia of the

car’s two rear wheels and axle is 0.24 kg-m2. The total

moment of inertia of the two front wheels is 0.2 kg-m2.

The radius of the tires is 0.3 m. The mass of the car,

including the wheels, is 1480 kg. The car is moving at

100 km/h. If the car’s brakes exert a constant retarding

couple of 650 N-m on each wheel and the tires do not

slip, what distance is required for the car to come to a

stop? (See Example 19.2.)

574

Problem 19.29 The radius of the pulley is R=

100 mm and its moment of inertia is I=0.1 kg-m2. The

mass m=5 kg. The spring constant is k=135 N/m.

The system is released from rest with the spring

unstretched. Determine how fast the mass is moving

when it has fallen 0.5 m.

R

km

Problem 19.30 The masses of the bar and disk are

14 kg and 9 kg, respectively. The system is released

1.2 m 0.3 m

Solution: The work done by the weights of the bar and disk as

they fall is

=6.72 kg-m2,

so the disk’s ﬁnal kinetic energy is

we obtain

Problem 19.31 The masses of the bar and disk are

14 kg and 9 kg, respectively. The system is released

from rest with the bar horizontal. Determine the angular

Problem 19.32 The 45-kg crate is pulled up the

inclined surface by the winch. The coefﬁcient of kinetic

friction between the crate and the surface is µk=0.4.

The moment of inertia of the drum on which the cable is

being wound is IA=4 kg-m2. The crate starts from rest,

and the motor exerts a constant couple M=50 N-m

on the drum. Use the principle of work and energy to

determine the magnitude of the velocity of the crate

when it has moved 1 m.

0.15 m

AM

20°

Solution: The normal force is

M

576

Problem 19.33 The 2-ft slender bars each weigh 4 lb,

and the rectangular plate weighs 20 lb. If the system is

released from rest in the position shown, what is the

velocity of the plate when the bars are vertical? 45°

is the change in height, from which U=14.06 ft-lb. The kinetic

energy is

T2=1

2Wplate

gv2+21

2WbarL2

3gω2=0.3523v2,

where ω=v

Lhas been used.

Substitute into U=T2and solve: v=6.32 ft/s.

v

Problem 19.34 The mass of the 2-m slender bar is

8 kg. A torsional spring exerts a counterclockwise cou-

ple kθ on the bar, where k=40 N-m/rad and θis in

radians. The bar is released from rest with θ=5◦. What

is the magnitude of the bar’s angular velocity when

θ=60◦?

k

u

Problem 19.35 The mass of the suspended object Ais

8 kg. The mass of the pulley is 5 kg, and its moment of

inertia is 0.036 kg-m2. If the force T=70N is applied

to the stationary system, what is the magnitude of the

velocity of Awhen it has risen 0.2 m?

T

0.4 m.

T1=0,V

1=0,T

2=1

Problem 19.36 The mass of the left pulley is 7 kg,

and its moment of inertia is 0.330 kg-m2. The mass of

the right pulley is 3 kg, and its moment of inertia is

0.054 kg-m2. If the system is released from rest, how

fast is the 18-kg mass moving when it has fallen 0.1 m?

0.3 m

0.2 m

9 kg

18 kg

A

B

Solution: When mass Cfalls a distance x, the center of pulley A

578

Problem 19.37 The 18-kg ladder is released from rest

with θ=10◦. The wall and ﬂoor are smooth. Modeling

the ladder as a slender bar, use conservation of energy to

determine the angular velocity of the bar when θ=40◦.

θ

2. The angular velocity

about this center is ω=2

Lv, where vis the velocity of the center

of mass of the ladder. The kinetic energy of the ladder is