Engineering Mechanics - Dynamics
2 rr'' 2 r'
2 a b
2 cos T '
2 v r r' v T r T ' a r r'' r T '
2 a T 2 r' T ' v r
0 m s v T 12 ft s a r
216 ft s
2 a T 0 ft s
2 r'' a b
2 cos T '
2
2 r
2 r'
2
* Problem 12-172
If the end of the cable at A is pulled down with speed v , determine the speed at which block B rises.
Given: v 2 m s
Solution: v
A v
L 2 s
B s
A
0 2 v
B v
A v
B v
B v
A
2
1 m s
Chapter 12
Problem 12-173
If the end of the cable at A is pulled down with speed v , determine the speed at which block B rises.
Given: v 2 m s
121

Engineering Mechanics - Dynamics
Solution: v
A v
L
1 s
A
2 s
C
0 v
A
2 v
C v
C v
A
2
L
2 s
B s
C v
B v
C
2 s
B
0 2 v v
B
0.5
m s
B v
C
Problem 12-174
Determine the constant speed at which the cable at A must be drawn in by the motor in order to hoist the load at B a distance d in a time t .
Given: d 15 ft t 5 s
Solution:
L 4 s
B s
A
0 4 v
B v
A v
A
4 v
B v
A
4
© t d v
A
12 ft s
Problem 12-175
Determine the time needed for the load at B to attain speed v , starting from rest, if the cable is drawn into the motor with acceleration a .
Given: v 8 m s
122
Chapter 12

Engineering Mechanics - Dynamics a 0.2
m s
2
Solution: v
B v
L 4 s
B s
A t 160 s
* Problem 12–176
If the hydraulic cylinder at H draws rod BC in by a distance d , determine how far the slider at A moves.
Given: d 8 in
Solution:
' s
H d
L s
A
2 s
H
' s
A
2 ' s
H
0 ' s
A
2 ' s
H
' s
A
16 in
Problem 12-177
The crate is being lifted up the inclined plane using the motor M and the rope and pulley arrangement shown. Determine the speed at which the cable must be taken up by the motor in order to move the crate up the plane with constant speed v .
Given: v 4 ft s
123
Chapter 12

Engineering Mechanics - Dynamics
Solution: v
A v
L 2 s
A s
A s
P
0 3 v
A v
P
3 v
A v
P v
P
12 ft s
Problem 12-178
Determine the displacement of the block at B if A is pulled down a distance d .
Given: d 4 ft
Solution:
' s
A d
L
1
2 s
A
2 s
C
L
2 s
B s
C s
B
0 2 ' s
A
2 ' s
C
' s
C
' s
A
0 2 ' s
B
' s
B
' s
C
2
' s
C
' s
B
2 ft
Problem 12–179
The hoist is used to lift the load at D. If the end A of the chain is travelling downward at v
A
and the end B is travelling upward at v
B
, determine the velocity of the load at D .
Given: v
A
5 ft s v
B
2 ft s
124
Chapter 12

Engineering Mechanics - Dynamics
Solution:
L s
B s
A
2 s
D v
D v
B
2 v
A
0 v
B v
A
2 v
D v
D
1.5
ft s
Positive means down,
Negative means up
* Problem 12-180
The pulley arrangement shown is designed for hoisting materials. If BC remains fixed while the plunger P is pushed downward with speed v , determine the speed of the load at A .
Given: v 4 ft s
Solution: v
P v
L 6 s
P s
A v
A
6 v
P
0 6 v v
A
P v
A
24 ft s
Problem 12-181
If block A is moving downward with speed v
A
while C is moving up at speed v
C
, determine the speed of block B .
Given: v
A
4 ft s
125
Chapter 12

Engineering Mechanics - Dynamics v
C
Solution:
2 ft s
S
A
2 S
B
S
C
L
Taking time derivative: v
A
2 v
B v
C
0 v
B v
B
1 ft s v
C
2 v
A
Positive means down, negative means up.
Problem 12-182
If block A is moving downward at speed v
A
while block C is moving down at speed v
C
, determine the relative velocity of block B with respect to C .
Given: v
A
6 ft s v
C
18 ft s
Solution:
S
A
2 S
B
S
C
L
Taking time derivative v
A
2 v
B v
C
0 v
B v
A
2 v
C v
BC v
B v
C v
B v
BC
12 ft s
30 ft s
Positive means down, negative means up
Problem 12–183
The motor draws in the cable at C with a constant velocity v
C
. The motor draws in the cable at
D with a constant acceleration of a
D
. If v
D
= 0 when t = 0, determine (a) the time needed for block A to rise a distance h , and (b) the relative velocity of block A with respect to block B when this occurs.
126
Chapter 12

Engineering Mechanics - Dynamics
Given: v
C
4 m s a
D
8 m s
2 h 3 m
Solution:
L
1 s
D
2 s
A
0 v
D
2 v
A
0 a
D
L
2 s
B
2 a
A s
B s
C
0 2 v
B v
C
0 2 a
B a
C a
A a
D
2 v
A a
A t s
A h a
A
¨ t
2
2 t
2 h a
A t 1.225 s v
A a
A t v
B
1
2 v
C v
AB v
A v
B v
AB
2.90
m s
* Problem 12-184
If block A of the pulley system is moving downward with speed v
A
while block C is moving up at v
C
determine the speed of block B .
Given: v
A
4 ft s v
C
2 ft s
127
Chapter 12

Engineering Mechanics - Dynamics
Solution:
S
A
2 S
B
2 S
C
L v
A
2 v
B
2 v
C
0 v
B
2 v
C
2 v
A v
B
0 m s
Problem 12–185
If the point A on the cable is moving upwards at v
A
, determine the speed of block B .
Given: v
A
14 m s
Solution:
L
1 s
D s
A s
D s
E
0 2 v
D
L
2 s
D v
A s
E v
E s
C
0 v
D
L
3 s
C v
C s
D
2 v
E s
C s s
E
E
0 2 v
C v
D v
E
Guesses v
C
1 m s v
D
1 m s v
E
1 m s
Given 0 2 v
D
0 v
D v v
C
A v
E
2 v
E
0 2 v
C v
D v
E
§¨
¨ v
C v
D
·¸
¸ v
E
Find v
C v
D v
E
§¨
¨ v
C v
D
·¸
¸ v
E
¨
¨
©
2
10
6
¸
¸
¹ m s v
B v
C v
B
2 m s
Positive means down,
Negative means up
128
Chapter 12

Engineering Mechanics - Dynamics
Problem 12-186
The cylinder C is being lifted using the cable and pulley system shown. If point A on the cable is being drawn toward the drum with speed of v
A
, determine the speed of the cylinder.
Given: v
A
2 m s
Solution:
L 2 s
C s
C s
A
0 3 v
C v
A v
C v
A
3 v
C
0.667
m s
Positive means down, negative means up.
Problem 12–187
The cord is attached to the pin at C and passes over the two pulleys at A and D . The pulley at A is attached to the smooth collar that travels along the vertical rod. Determine the velocity and acceleration of the end of the cord at B if at the instant s
A
= b the collar is moving upwards at speed v , which is decreasing at rate a .
Given: a b
3 ft
4 ft v
A
5 ft s a
A
2 ft s
2
Solution:
L 2 a
2
Guesses
Given s
A
2 s
B s
A b v
B
0
1 ft s
2 s
A v
A a
2 s
A
2 a
B
1 ft s
2 v
B
129
Chapter 12