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14:440:222 Dynamics
(Lecture Note #5)
Instructor: Prof. Peng Song
Rutgers University
Today’s Lecture
Kinematic Analysis of TwoParticle Systems
• Dependent motion
• Relative motion
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View Full Document Applications
The cable and pulley system shown
can be used to modify the speed of
the mine car, A, relative to the speed
of the motor, M.
It is important to establish the
relationships between the various
motions in order to determine the
power requirements for the motor
and the tension in the cable.
For instance, if the speed of the cable (P) is known because we
know the motor characteristics, how can we determine the speed
of the mine car?
Will the slope of the track have any impact on
the answer?
Rope and pulley arrangements are
often used to assist in lifting
heavy objects.
The total lifting
force required from the truck
depends on both the weight and
the acceleration of the cabinet.
How can we determine the
acceleration and velocity of the
cabinet if the acceleration of
the truck is known?
Applications (cont’d)
Dependent Motion
In many kinematics problems, the motion of one object will depend
on the motion of another object.
The motion of each block can be related mathematically by defining
position coordinates, s
A
and s
B
.
Each coordinate axis is defined
from a fixed point or datum line, measured positive along each
plane in the direction of motion of each block.
The blocks in this figure are
connected by an inextensible cord
wrapped around a pulley. If block
A moves downward along the
inclined plane, block B will move
up the other incline.
In this example, position
coordinates s
A
and s
B
can be
defined from fixed datum lines
extending from the center of
the pulley along each incline to
blocks A and B.
If the cord has a fixed length, the position coordinates s
A
and
s
B
are related mathematically by the equation
s
A
+ l
CD
+ s
B
= l
T
Here l
T
is the total cord length and l
CD
is the length of cord
passing over the arc CD on the pulley.
Dependent Motion (cont’d)
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View Full Document The negative sign indicates that as A moves down the incline
(positive s
A
direction), B moves up the incline (negative s
B
direction).
Accelerations can be found by differentiating the velocity expression.
Prove to yourself that a
B
= a
A
.
ds
A
/dt + ds
B
/dt = 0
=>
v
B
= v
A
The velocities of blocks A and B
can be related by differentiating the
position equation.
Note that l
CD
and l
T
remain constant, so
dl
CD
/dt =
dl
T
/dt = 0
Dependent Motion (cont’d)
Dependent Motion Example
Consider a more complicated
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This document was uploaded on 10/31/2011 for the course ENG MECH 232 at Rutgers.
 Fall '11
 R.C.Hibbeler

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