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lecture+notes+5 - 14:440:222 Dynamics(Lecture Note#5...

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14:440:222 Dynamics (Lecture Note #5) Instructor: Prof. Peng Song Rutgers University Today’s Lecture Kinematic Analysis of Two-Particle Systems Dependent motion Relative motion
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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)
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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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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 .
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