M What is the moment of inertia of the pulley? The blocks are initially held at rest, then released. At the instant the larger mass M has dropped 1.5 m, what is the speed of each block? Please remember to draw free-body diagrams. What is the tension in the rope at that same instant? E
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The free body diagram of the system is :
Due to the moment of inertia of the pulley , the tension and will not be the same . The change in the tension on either side of the pulley , will provide an torque to the pulley , which is calculated by using the formula , , where is the moment of inertia of a solid disk , is the mass of pulley , is the radius of the pulley , is the angular acceleration of the pulley and a is the acceleration of the system . Thus ,
According to the Newton's second law , the net force acting on block is , where the forces acting are the torque upward and weight force downward , which is , . Thus ,
According to the Newton's second law , the net force acting on block is , where the forces acting are the torque upward and weight force downward , which is , . Thus ,
From (2) and (3) equation (1) becomes
According to the law of conservation of energy , energy can neither be created nor destroyed , but can only be converted from one form to another . Thus the initial energy in the system which is the potential energy of the two blocks , will be converted to their final potential energy after dropping a height of , their kinetic energy and the rotational energy of the pulley , which is , , where is the angular velocity and v is the velocity of the system .Thus ,
, where is the initial height of block , is the initial height of block , is the final height of block , is the final height of block and v is the final velocity of the blocks.
Calculate moment of inertia of the pulley by using the formula , where is the mass of pulley and is the radius of the pulley .
Calculate the speed of the blocks by using relation (5) , which is , , where is the acceleration due to gravity , is the change in height , is the mass of pulley , and .
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