Mechanical Engineering Chapter 19 Problem The Slender Bar Released From Rest With The Horizontal Surface Smoothwhat

Problem 19.38 The 8-kg slender bar is released from

rest with θ=60◦. The horizontal surface is smooth.

What is the bar’s angular velocity when θ=30◦.

2 m

Solution: The bar’s potential energy is

Problem 19.39 The mass and length of the bar are

m=4 kg and l=1.2 m. The spring constant is k=

180 N/m. If the bar is released from rest in the position

θ=10◦, what is its angular velocity when it has fallen

to θ=20◦?

k

Solution: If the spring is unstretched when θ=0, the stretch of

the spring is

580

Problem 19.40 The 4-kg slender bar is pinned to a

2-kg slider at Aand to a 4-kg homogenous cylindrical

disk at B. Neglect the friction force on the slider and

assume that the disk rolls. If the system is released from

rest with θ=60◦, what is the bar’s angular velocity

when θ=0? (See Example 19.3.) 1 m

A

θ

Problem 19.41* The sleeve Pslides on the smooth

horizontal bar. The mass of each bar is 4 kg and the

mass of the sleeve Pis 2 kg. If the system is released

from rest with θ=60◦, what is the magnitude of the

velocity of the sleeve Pwhen θ=40◦?

1.2 m

O

Q

P

θ

1.2 m

582

Problem 19.42* The system is in equilibrium in the

position shown. The mass of the slender bar ABC is

6 kg, the mass of the slender bar BD is 3 kg, and the

mass of the slider at Cis 1 kg. The spring constant is

k=200 N/m. If a constant 100-N downward force is

applied at A, what is the angular velocity of the bar

ABC when it has rotated 20◦from its initial position?

A

1 m

B

1 m

Solution: Choose a coordinate system with the origin at Dand

Problem 19.43* The masses of bars AB and BC are

5 kg and 3 kg, respectively. If the system is released

from rest in the position shown, what are the angular

velocities of the bars at the instant before the joint B

hits the smooth ﬂoor? 1 m

2 m 1 m

AB

C

Solution: The work done by the weights of the bars as they fall

is

y

A

584

Problem 19.44* Bar AB weighs 5 lb. Each of the

sleeves Aand Bweighs 2 lb. The system is released

from rest in the position shown. What is the magnitude

of the angular velocity of the bar when sleeve B has

moved 3 in to the right?

9 in

4 in 12 in

y

x

B

A

Solution: Find the geometry in position 2

Problem 19.45* Each bar has a mass of 8 kg and a

length of l m. The spring constant is k=100 N/m, and

the spring is unstretched when θ=0. If the system is

released from rest with the bars vertical, what is the mag-

nitude of the angular velocity of the bars when

θ=30◦?

θ

k

586

Problem 19.46* The system starts from rest with the

crank AB vertical. A constant couple Mexerted on the

crank causes it to rotate in the clockwise direction, com-

pressing the gas in the cylinder. Let sbe the displace-

ment (in meters) of the piston to the right relative to

its initial position. The net force toward the left exerted

on the piston by atmospheric pressure and the gas in

the cylinder is 350/(1−10s) N. The moment of inertia

of the crank about Ais 0.0003 kg-m2. The mass of the

connecting rod BC is 0.36 kg, and the center of mass of

the rod is at its midpoint. The connecting rod’s moment

of inertia about its center of mass is 0.0004 kg-m2. The

mass of the piston is 4.6 kg. If the clockwise angular

velocity of the crank AB is 200 rad/s when it has rotated

90◦from its initial position, what is M? (Neglect the

50 mm

A

M

B

C

125 mm

Problem 19.47* In Problem 19.46, if the system starts

from rest with the crank AB vertical and the couple

M=40 N-m, what is the clockwise angular velocity of

AB when it has rotated 45◦from its initial position?

Problem 19.48 The moment of inertia of the disk

about Ois 22 kg-m2.Att=0, the stationary disk is

subjected to a constant 50 N-m torque.

(a) Determine the angular impulse exerted on the disk

from t=0tot=5s.

(b) What is the disk’s angular velocity at t=5s?

50 N-m

O

Problem 19.49 The moment of inertia of the jet

engine’s rotating assembly is 400 kg-m2. The assembly

starts from rest. At t=0, the engine’s turbine exerts

a couple on it that is given as a function of time by

M=6500 −125tN-m.

(a) What is the magnitude of the angular impulse

exerted on the assembly from t=0tot=20 s?

(b) What is the magnitude of the angular velocity of

the assembly (in rpm) at t=20 s?

588

Problem 19.50 An astronaut ﬁres a thruster of his

maneuvering unit, exerting a force T=2(1+t)N,

where tis in seconds. The combined mass of the

astronaut and his equipment is 122 kg, and the moment

of inertia about their center of mass is 45 kg-m2.

Modeling the astronaut and his equipment as a

rigid body, use the principle of angular impulse and

momentum to determine how long it takes for his angular

velocity to reach 0.1 rad/s. 300 mm

T

Solution: From the principle of impulse and angular momentum,

Problem 19.51 The combined mass of the astronaut

and his equipment is 122 kg, and the moment of inertia

about their center of mass is 45 kg-m2. The maneuvering

unit exerts an impulsive force Tof 0.2-s duration, giving

him a counterclockwise angular velocity of 1 rpm.

(a) What is the average magnitude of the impulsive

force?

(b) What is the magnitude of the resulting change in

the velocity of the astronaut’s center of mass?

T

300 mm

Solution:

(a) From the principle of moment impulse and angular momentum,

590

Problem 19.52 Aﬂywheel attached to an electric

motor is initially at rest. At t=0, the motor exerts

a couple M=200e−0.1tN-m on the ﬂywheel. The

moment of inertia of the ﬂywheel is 10 kg-m2.

(a) What is the ﬂywheel’s angular velocity at t=10 s?

(b) What maximum angular velocity will the ﬂywheel

attain?

Problem 19.53 A main landing gear wheel of a Boeing

777 has a radius of 0.62 m and its moment of inertia

is 24 kg-m2. After the plane lands at 75 m/s, the skid

marks of the wheel’s tire is measured and determined to

be 18 m in length. Determine the average friction force

exerted on the wheel by the runway. Assume that the

airplane’s velocity is constant during the time the tire

skids (slips) on the runway.

Problem 19.54 The force a club exerts on a 0.045-kg

golf ball is shown. The ball is 42 mm in diameter and

can be modeled as a homogeneous sphere. The club is in

contact with the ball for 0.0006 s, and the magnitude of

the velocity of the ball’s center of mass after the ball is

hit is 36 m/s. What is the magnitude of the ball’s angular

velocity after it is hit?

2.5 mm

F

Problem 19.55 Disk Ainitially has a counterclock-

wise angular velocity ω0=50 rad/s. Disks Band Care

initially stationary. At t=0, disk Ais moved into con-

tact with disk B. Determine the angular velocities of

19.4.)

0.2 m

0.4 m 0.3 m

v0

Solution: The FBDs

Given:

mA=4kg,r

A=0.2 m,I

A=1/2mArA2=0.08 kg-m2

F2

F1

592

Problem 19.56 In Example 19.5, suppose that in a

second test at a higher velocity the angular velocity

of the pole following the impact is ω=0.81 rad/s, the

horizontal velocity of its center of mass is v=7.3

m/s, and the duration of the impact is t =0.009 s.

Determine the magnitude of the average force the car

exerts on the pole in shearing off the supporting bolts.

Do so by applying the principle of angular impulse and

momentum in the form given by Eq. (19.32).

Solution: Using the data from Example 19.5 we write the linear

and angular impulse momentum equations for the pole. Fis the force

of the car and Sis the shear force in the bolts

(F −S)t =mv

Problem 19.57 The force exerted on the cue ball by

the cue is horizontal. Determine the value of hfor which

the ball rolls without slipping. (Assume that the frictional

force exerted on the ball by the table is negligible.)

h

R

Solution: From the principle of moment impulse and angular

594

Problem 19.58 In Example 19.6, we neglected the

moments of inertia of the two masses mabout the axes

through their centers of mass in calculating the total

angular momentum of the person, platform, and masses.

Suppose that the moment of inertia of each mass about

the vertical axis through its center of mass is IM=

0.001 kg-m2. If the person’s angular velocity with her

arms extended to r1=0.6misω1=1 revolution per

second, what is her angular velocity ω2when she pulls

the masses inward to r2=0.2 m? Compare your result

r1

r2

Solution: Using the numbers from Example 19.6, we conserve

angular momentum

HO1=(IP+2mr2

1+2IM)ω1

=(0.4 kg-m2+2[4 kg][0.6m]

2+2[0.001 kg-m2])1rev

s

HO2=(IP+2mr2

2+2IM)ω2

s.

Problem 19.59 Two gravity research satellites (mA=

250 kg, IA=350 kg-m2;mB=50 kg, IB=16 kg-m2)

are tethered by a cable. The satellites and cable

rotate with angular velocity ω0=0.25 rpm. Ground

controllers order satellite Ato slowly unreel 6 m of

additional cable. What is the angular velocity afterward?

A

B

0

ω

Problem 19.60 The 2-kg bar rotates in the horizontal

plane about the smooth pin. The 6-kg collar Aslides on

the smooth bar. Assume that the moment of inertia of

the collar Aabout its center of mass is negligible; that

is, treat the collar as a particle. At the instant shown, the

angular velocity of the bar is ω0=60 rpm and the dis-

tance from the pin to the collar is r=1.8 m. Determine

the bar’s angular velocity when r=2.4m.

r

A

3 m

k

v0

Problem 19.61 The 2-kg bar rotates in the horizontal

plane about the smooth pin. The 6-kg collar Aslides

Solution: Angular momentum is conserved

1

Problem 19.62* The 2-kg bar rotates in the horizontal

plane about the smooth pin. The 6-kg collar Aslides

on the smooth bar. The moment of inertia of the collar

r=2.4m.

Solution: Angular momentum is conserved

1

T1=1

21

3(2kg)(3m)2+(6kg)(1.8 m)2+(0.2 kg-m2)(2 rad/s)2

596

Problem 19.63 The circular bar is welded to the verti-

cal shafts, which can rotate freely in bearings at Aand B.

Let Ibe the moment of inertia of the circular bar and

shafts about the vertical axis. The circular bar has an

A

Problem 19.64 The 10-lb bar is released from rest

in the 45◦position shown. It falls and the end of the

bar strikes the horizontal surface at P. The coefﬁcient

of restitution of the impact is e=0.6. When the bar

rebounds, through what angle relative to the horizontal

will it rotate?

3 ft

45⬚

P

Solution: We solve the problem in three phases.

Problem 19.65 The 10-lb bar is released from rest in

the 45◦position shown. It falls and the end of the bar

strikes the horizontal surface at P. The bar rebounds to

a position 10◦relative to the horizontal. If the duration

of the impact is 0.01 s, what is the magnitude of the

average vertical force the horizontal surface exerted on

the bar at P?

3 ft

45⬚

P

598

Problem 19.66 The 4-kg bar is released from rest in

the horizontal position above the ﬁxed projection at A.

The distance b=0.35 m. The impact of the bar with the

projection is plastic; that is, the coefﬁcient of restitution

of the impact is e=0. What is the bar’s angular velocity

immediately after the impact?

1 m

0.2 m

b

A

Problem 19.67 The 4-kg bar is released from rest in

the bar’s center of mass to be zero immediately after the

impact? What is the bar’s angular velocity immediately

after the impact?

1 m

0.2 m

b

A