COSMOS: Complete Online Solutions Manual Organization System

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P. Beer, E. Russell Johnston, Jr.,
Elliot R. Eisenberg, William E. Clausen, David Mazurek, Phillip J. Cornwell
2007 The McGraw-Hill Companies.

Chapter 18, Solution 155.


Angular velocity of shaft DE and arm CBA:
2
= i
Angular velocity of disk A:
2 1
= + i j
Angular velocity about its mass center A:
2 2
2 1 2 1
1 1
4 2
A x x y y z z x y
I I I I I mr mr = + + = + = + H i j k i j i j
Let the reference frame Oxyz be rotating with angular velocity
2
. = i


( )
2 2
2 1 2
1 1
4 2
A A A A
Oxyz
mr mr = + = + + H H H i j i H
! !
! !

2 2 2
2 1 1 2
1 1 1
4 2 2
mr mr mr = + + i j k ! !
Velocity of point A: ( )
2 / 2 2 2 A AO
b c b c = = + = + v i r i k j j k
Acceleration of point A:
2 / 2 A AO A
= + a j r j v !
( )
( )
2
2 2 2 2 A
b c c b = + + a j k ! !

Consider the system of particles consisting of the shaft, the arm, and the disk. Neglect the mass of the arm.
A
m = F a
y z y z A
D D E E m + + + = j k j k a

COSMOS: Complete Online Solutions Manual Organization System

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P. Beer, E. Russell Johnston, Jr.,
Elliot R. Eisenberg, William E. Clausen, David Mazurek, Phillip J. Cornwell
2007 The McGraw-Hill Companies.


Resolve into components.
( )
2
2 2 y y
D E m b c + = !
( )
2
2 2 z z
D E m c b + = + !
/ D D A A D A
m = = + M H H r a
! !

( ) ( ) ( ) 2
O y z A A
M l E E l c b m + + = + + i i j k H i j k a
!

( )
2 2 2 2 2
2 1 2 2
1 1
2 2
4 2
O z y
M lE lE m r b c m r lc lb
| | | |
+ = + + +
| |
\ . \ .
i j k i j ! ! !

2 2
1 2 2 2
1
2
m r lb lc
| |
+ +
|
\ .
k !

2 2 2
2
1
:
4
O
M m r b c
| |
= + +
|
\ .
i !

2 2 2 2
1 2 2 2 1 2 2 2
1 1
: ,
2 2 2 2
y y
m m
E r lb lc D r lb lc
l l

| | | |
= + = +
| |
\ . \ .
k ! !

2 2 2 2
2 2 1 2 2 1
1 1
: ,
2 2 2 2
z z
m m
E lc lb r D lc lb r
l l

| | | |
= + = + +
| |
\ . \ .
j ! ! ! !
Data: 3 kg, 0.2 m, 0.225 m, 0.3 m m r b c l = = = = =
2
1 1 2 2
16 rad/s, 0, 8 rad/s, 6 rad/s = = = = ! !
(a) ( ) ( ) ( ) ( ) ( )
2 2 2 1
3 0.2 0.225 0.225 6
4
O
M
(
= + +
(


( ) 2.00 N m
O
= M i !
(b)
( )( )
( ) ( )( ) ( )( )( ) ( )( )( )
2 2 3 1
0.2 16 8 0.3 0.225 6 0.3 0.225 8 32.4 N
2 0.3 2
y
D
(
= + =
(




( )( )
( )( )( ) ( )( )( )
2 3
0.3 0.225 6 0.3 0.225 8 0 23.6 N
2 0.3
z
D
(
= + + =
(


( ) ( ) 32.4 N 23.6 N = + D j k !
( )( )
( ) ( )( ) ( )( )( ) ( )( )( )
2 2 3 1
0.2 16 8 0.3 0.225 6 0.3 0.225 8 6.78 N
2 0.3 2
y
E
(
= + =
(




( )( )
( )( )( ) ( )( )( )
2 3
0.3 0.225 6 0.3 0.225 8 0 23.6 N
2 0.3
z
E
(
= + + =
(


( ) ( ) 6.78 N 23.6 N = + E j k !