Mechanical Engineering Chapter 21 Problem Active Example Suppose Thatr The Pulley Has Radius And Its Momentof

Problem 21.1 In Active Example 21.1, suppose that

the pulley has radius R=100 mm and its moment

of inertia is I=0.005 kg-m2. The mass m=2 kg,

and the spring constant is k=200 N/m. If the mass

is displaced downward from its equilibrium position

and released, what are the period and frequency of the

resulting vibration?

R

km

m+I

R2



(2kg)+(0.005 kg-m2)

(0.1m)2

Thus

Problem 21.2 In Active Example 21.1, suppose that

the pulley has radius R=4 in and its moment of inertia

is I=0.06 slug-ft2. The suspended object weighs 5 lb,

and the spring constant is k=10 lb/ft. The system is

initially at rest in its equilibrium position. At t=0, the

suspended object is given a downward velocity of 1 ft/s.

Determine the position of the suspended object relative

to its equilibrium position as a function of time.

R

km

Problem 21.3 The mass m=4 kg. The spring is un-

stretched when x=0. The period of vibration of the

mass is measured and determined to be 0.5 s. The mass

is displaced to the position x=0.1 m and released from

x

Problem 21.4 The mass m=4 kg. The spring is un-

stretched when x=0. The frequency of vibration of the

mass is measured and determined to be 6 Hz. The mass

is displaced to the position x=0.1 m and given a veloc-

x

37.7 rad/s =0.133 m.

728

Problem 21.5 The mass m=4 kg and the spring con-

stant is k=64 N/m. For vibration of the spring-mass

oscillator relative to its equilibrium position, determine

x=0. At t=0,x =0 and the mass has a velocity of

2 m/s down the inclined surface. What is the value of x

at t=0.8s?

md2x

dt2+kx =mg sin 20◦⇒d2x

dt2+k

mx=gsin 20◦

Putting in the numbers we have

Problem 21.7 Suppose that in a mechanical design

course you are asked to design a pendulum clock, and

you begin with the pendulum. The mass of the disk is

2 kg. Determine the length Lof the bar so that the period

Problem 21.8 The mass of the disk is 2 kg and the

mass of the slender bar is 0.4 kg. Determine the length

Lof the bar so that the period of small oscillations of

the pendulum is 1 s.

Strategy: Draw a graph of the value of the period for

a range of lengths Lto estimate the value of Lcorre-

sponding to a period of 1 s.

50 mm

L

Problem 21.9 The spring constant is k=785 N/m.

The spring is unstretched when x=0. Neglect the mass

of the pulley, that is, assume that the tension in the rope

is the same on both sides of the pulley. The system is

released from rest with x=0. Determine xas a function

730

Problem 21.10 The spring constant is k=785 N/m.

x=0. Determine xas a function of time.

Solution: Let T1be the tension in the rope on the left side, and

Problem 21.11 A “bungee jumper” who weighs 160

lb leaps from a bridge above a river. The bungee cord

has an unstretched length of 60 ft, and it stretches an

additional 40 ft before the jumper rebounds. Model the

cord as a linear spring. When his motion has nearly

stopped, what are the period and frequency of his vertical

oscillations? (You can’t model the cord as a linear spring

during the early part of his motion. Why not?)

f=3.13 s

Problem 21.12 The spring constant is k=800 N/m,

Solution: The equations of motion are

(b) The solution to the differential equation is

Problem 21.13 The spring constant is k=800 N/m,

and the spring is unstretched when x=0. The mass of

each object is 30 kg. The inclined surface is smooth. The

Solution: Let T1be the tension in the rope on the left of the

pulley, and T2be the tension in the rope on the right of the pulley.

The equations of motion are

3.59 rad/s =0.279 m.

732

Problem 21.14 The 20-lb disk rolls on the horizontal

surface. Its radius is R=6 in. Determine the spring con-

stant kso that the frequency of vibration of the system

relative to its equilibrium position is f=1 Hz.

kR

Problem 21.15 The 20-lb disk rolls on the horizontal

surface. Its radius is R=6 in. The spring constant is

k=15 lb/ft. At t=0, the spring is unstretched and the

disk has a clockwise angular velocity of 2 rad/s. What

is the amplitude of the resulting vibrations of the center

of the disk?

Problem 21.16 The 2-lb bar is pinned to the 5-lb disk.

The disk rolls on the circular surface. What is the fre-

quency of small vibrations of the system relative to its

vertical equilibrium position?

15 in

4 in

Solution: Use energy methods.

Problem 21.17 The mass of the suspended object A

is 4 kg. The mass of the pulley is 2 kg and its moment

of inertia is 0.018 N-m2. For vibration of the system

relative to its equilibrium position, determine (a) the fre-

quency in Hz and (b) the period.

k

Solution: Use energy methods

734

Problem 21.18 The mass of the suspended object Ais

x=0. At t=0, the system is released from rest with

x=0. What is the velocity of the object Aat t=1s?

Problem 21.19 The thin rectangular plate is attached

to the rectangular frame by pins. The frame rotates

with constant angular velocity ω0=6 rad/s.The angle β

between the zaxis of the body-ﬁxed coordinate system

z

b

v0

Problem 21.20 Consider the system described in

Problem 21.19. At t=0, the angle β=0.01 rad and

dβ/dt =0. Determine βas a function of time.

Problem 21.21 A slender bar of mass mand length l

is pinned to a ﬁxed support as shown. A torsional

spring of constant kattached to the bar at the support

is unstretched when the bar is vertical. Show that the

equation governing small vibrations of the bar from its

dt2+ω2θ=0,where ω2=(k −1

2mgl)

1

3ml2.

θ

2Idθ

dt 2

The potential energy is the sum of the energy in the spring and the

gravitational energy associated with the change in height of the center

of mass of the bar,

For small amplitude vibrations sin θ→θ, and the canonical form (see

Eq. (21.4)) of the equation of motion is

k−mgL

736

Problem 21.22 The initial conditions of the slender

bar in Problem 21.21 are

t=0θ=0

dθ

dt =˙

θ0.

(a) If k>1

2mgl, show that θis given as a function of

time by

θ=˙

θ0

ωsin ωt, where ω2=(k −1

2mgl)

1

3ml2.

(b) If k<1

2mgl, show that θis given as a function of

time by

θ=˙

θ0

2h(eht −e−ht ), where h2=(1

2mgl −k)

1

3ml2.

Problem 21.23 Engineers use the device shown to

measure an astronaut’s moment of inertia. The horizontal

board is pinned at Oand supported by the linear spring

with constant k=12 kN/m. When the astronaut is not

present, the frequency of small vibrations of the board

about Ois measured and determined to be 6.0 Hz.

When the astronaut is lying on the board as shown, the

frequency of small vibrations of the board about Ois

2.8 Hz. What is the astronaut’s moment of inertia about

the zaxis?

y’

x’

x

y

1.90 m

O

Problem 21.24 In Problem 21.23, the astronaut’s cen-

ter of mass is at x=1.01 m, y=0.16 m, and his mass

is 81.6 kg. What is his moment of inertia about the z′

axis through his center of mass?

=24.2 kg-m2.

738

c

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Problem 21.25* Aﬂoating sonobuoy (sound-measur-

ing device) is in equilibrium in the vertical position

shown. (Its center of mass is low enough that it is stable

in this position.) The device is a 10-kg cylinder 1 m

in length and 125 mm in diameter. The water density

is 1025 kg/m3, and the buoyancy force supporting the

buoy equals the weight of the water that would occupy

the volume of the part of the cylinder below the surface.

If you push the sonobuoy slightly deeper and release it,

what is the frequency of the resulting vertical vibrations?

Problem 21.26 The disk rotates in the horizontal plane

with constant angular velocity =12 rad/s. The mass

m=2 kg slides in a smooth slot in the disk and is

attached to a spring with constant k=860 N/m. The

radial position of the mass when the spring is unstretched

is r=0.2m.

(a) Determine the “equilibrium” position of the mass,

the value of rat which it will remain stationary

relative to the center of the disk.

(b) What is the frequency of vibration of the mass re-

lative to its equilibrium position?

k

m

r

V

Problem 21.27 The disk rotates in the horizontal plane

with constant angular velocity =12 rad/s. The mass

m=2 kg slides in a smooth slot in the disk and is

0.4 m and dr/dt =0. Determine the position ras a

function of time.

Solution: Using polar coordinates, Newton’s second law in the r

direction is

¨r+(16.9 rad/s)2r=86 m/s2.

740

Problem 21.28 A homogeneous 100-lb disk with

radius R=1 ft is attached to two identical cylindrical

steel bars of length L=1 ft. The relation between the

moment Mexerted on the disk by one of the bars and

the angle of rotation, θ, of the disk is

disk is to be 10 Hz.

R

θ

∂θ =2GJ

L.

Problem 21.29 The moments of inertia of gears Aand

Bare IA=0.025 kg-m2and IB=0.100 kg-m2. Gear A

is connected to a torsional spring with constant k=

10 N-m/rad. What is the frequency of small angular

vibrations of the gears?

Problem 21.30 At t=0, the torsional spring in Prob-

lem 21.29 is unstretched and gear Bhas a counterclock-

wise angular velocity of 2 rad/s. Determine the coun-

terclockwise angular position of gear Brelative to its

equilibrium position as a function of time.

M=11.62 rad/s.

Assume a solution of the form θA=Asin ωt +Bcos ωt. Apply the

initial conditions,

742

Problem 21.31 Each 2-kg slender bar is 1 m in length.

What are the period and frequency of small vibrations

of the system?

Problem 21.32* The masses of the slender bar and

the homogeneous disk are mand md, respectively. The

spring is unstretched when θ=0. Assume that the disk

rolls on the horizontal surface.

(a) Show that the motion of the system is governed by

the equation

1

3+3md

2mcos2θd2θ

dt2−3md

2msin θcos θdθ

dt 2

−g

2lsin θ+k

m(1−cos θ)sin θ=0.

(b) If the system is in equilibrium at the angle θ=θe

and ˜

θ=θ−θeshow that the equation governing

small vibrations relative to the equilibrium posi-

tion is

1

3+3md

2mcos2θed2˜

θ

dt2+k

m(cos θe−cos2θe

+sin2θe)−g

2lcos θe˜

θ=0.

k

l

R

θ

Solution: (See Example 21.2.) The system is conservative.

from the center of mass to the center of rotation is

r=L−L

sin2θ+L−L

cos2θ=L

θ

L

k

−g

2Lsin θ+k

m(1−cos θ)sin θ=0,

744

Problem 21.33* The masses of the bar and disk

in Problem 21.32 are m=2 kg and md=4 kg,

respectively. The dimensions l=1 m and R=0.28 m,

and the spring constant is k=70 N/m.

(a) Determine the angle θeat which the system is in

equilibrium.

(b) The system is at rest in the equilibrium position,

and the disk is given a clockwise angular velocity

of 0.1 rad/s. Determine θas a function of time.

Solution:

746