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Two very interesting machine dynamics problems involve a pulley and a block, connected together by a cable that is wound around the outside of the pulley. This assembly forms a “simple hoist”. In one problem, the block is released from rest, causing the pulley to rotate, and in the other problem, a torque is applied to the pulley, causing the block to be raised.
As in all dynamics problems, start with free-body diagrams of each body, here the pulley and the block. Then apply the equations of motion to both bodies. From the pulley we get 3 equations and from the block 1 equation. As is the case in most dynamics problems, we are typically short 1 equation. This is where the kinematics of the problem are applied. For both these problems, the kinematics equation is that the acceleration of the block must equal the tangential acceleration of the tangent point on the pulley. At this point, we will have 3 equations in 3 unknowns. Solving these simultaneously gives the angular acceleration of the pulley, the acceleration of the block, and the tension in the cable.
For the problem where the block is released from rest, it turns out the tension in the cable is less than the value you would obtain from Statics, meaning the weight of the block. For the problem where a torque is applied to raise the block, the tension in the cable is greater than the weight of the block. This is where the biggest mistake is made in this type problem, setting the value of the tension in the cable to the weight of the block.
These two problems are indeed basic to solving machine design problems. Start with free-body diagrams each element of a machine. Then apply Newton’s Laws of Motion to each free-body diagram. if at that point you have more unknowns than equations, look to kinematic equations to obtain the necessary equations. Then solve the resulting equations for the unknowns. It was learning this process that for me dynamics became a topic i enjoyed instead of dreaded. – Dr. Tom
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