978-0073380308 Chapter 7 Solution Manual Part 18

which can be rearranged to obtain the following equation of motion

G

Computer Solution

This equation of motion can now be integrated using

mD200 kg

,

rD0:8

m,

kGD0:65

m,

D38ı

,

FindRoot!x‘!t“!0#.Motion,$t, 5.%“

which yields the following result for the period of oscillation

Period of Oscillation D5:12 s:

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1587

Problem 7.88

The uniform bar

AB

of mass

m

and length

L

is leaning against the corner with

0

when end

B

is given a slight nudge so that end

A

starts sliding down the wall as

B

slides along the ﬂoor. Assuming that friction is negligible between the bar and the two

surfaces against which it is sliding, determine the angle



at which end

A

will lose

contact with the vertical wall.

Solution

The FBD of the bar as it is sliding down the wall is shown at the right.

Balance Principles.

The Newton-Euler equations corresponding to this FBD

are:

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permission of McGraw-Hill, is prohibited.

Expanding cross products and equating coefficients, we obtain

2˛AB sin L

2!2

Computation.

Substituting the expressions for the mass moment of inertia,

aGx

, and

aGy

into the Newton-

Euler equations, we obtain the following system of three equations in the three unknowns

NA

,

NB

, and

˛AB

Solving the first equation for

NA

, the second for

NB

, and then substituting those results into the third equation,

The value

s

for which the bar separates from the wall is the value of



at which the force

NA

becomes

equal to zero. We can see from Eq. (1) that we need to know

˛AB

and

!AB

as functions of



before we can

find this angle. Having just found ˛AB , we can integrate it using the chain rule to find !AB as follows

d

d!AB

3g

)!2

AB D3g

L.1 cos /:

Substituting the expressions for !2

AB and ˛AB into Eq. (1) and setting the result equal to zero, we obtain

The values of



for which

sin D0

do not make physical sense in this context and the only value of



for

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1589

Problem 7.89

The uniform thin bar, which is leaning on the incline, is released

from rest in the position shown and slides in the vertical plane. The

contacts between the bar and the surface at ends

A

and

B

have

negligible friction. Determine the angular acceleration of the bar

immediately after it is released. Evaluate your answer for

mD3kg

,

LD0:75 m, D45ı, and D30ı.

Solution

The FBD of the bar immediately after it is released is shown on the

right, where

NA

is perpendicular to the inclined surface and

NB

is

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permission of McGraw-Hill, is prohibited.

which, when equating components, gives

cos 

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1591

Problem 7.90

The uniform ball of radius



and mass

m

is gently placed in the

bowl

B

with inner radius

R

and is released. The angle



measures

the position of the center of the ball at

G

with respect to a vertical

line through

O

. Assume that the system lies in the vertical plane.

Hint: In working the following problems, we recommend using

the r coordinate system shown.

Assuming that the ball rolls without slip, determine the acceler-

ation of the center of the ball at

G

, the angular acceleration of the

ball, and the force on the ball due to the bowl immediately after

the ball is released.

IGD2

5m2:

Force Laws. All forces have been accounted for on the FBD.

Kinematic Equations.

Because the center of the ball is in circular motion about point

O

, the acceleration

of point Gcan be written as

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permission of McGraw-Hill, is prohibited.

Computation.

Substituting the kinematic equations into the Newton-Euler equations yields the following

system of three equations in the three unknowns N,F, and R



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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1593

Problem 7.91

The uniform ball of radius



and mass

m

is gently placed in the

bowl

B

with inner radius

R

and is released. The angle



measures

the position of the center of the ball at

G

with respect to a vertical

line through

O

. Assume that the system lies in the vertical plane.

Hint: In working the following problems, we recommend using

the r coordinate system shown.

Assuming that the ball rolls without slip, that it weighs

3lb

,

is at the position

D40ı

, and is moving clockwise at

10 ft=s

,

determine the acceleration of the center of the ball at

G

and the

normal and friction force between the ball and the bowl. Use

RD4ft and D1:2 ft.

Solution

The FBD of the sphere as it rolls without slipping in the bowl is shown

on the right.

Balance Principles.

The Newton-Euler equations corresponding to

this FBD are:

XFrWmg cos NDmaGr ;

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permission of McGraw-Hill, is prohibited.

Since the ball rolls without slip on a stationary surface, aQ D0, and so

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1595

Problem 7.92

The uniform ball of radius



and mass

m

is gently placed in the

bowl

B

with inner radius

R

and is released. The angle



measures

the position of the center of the ball at

G

with respect to a vertical

line through

O

. Assume that the system lies in the vertical plane.

Hint: In working the following problems, we recommend using

the r coordinate system shown.

Assuming that friction is sufficient to prevent slipping, derive

the equation(s) of motion of the ball in terms of the angle .

IGD2

5m2:

Force Laws. All forces have been accounted for on the FBD.

Kinematic Equations.

Because the center of the ball is in circular motion about point

O

, the acceleration

R

Computation. Substituting the kinematic equations into the Newton-Euler equations gives

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permission of McGraw-Hill, is prohibited.