One end of a light inextensible string is fixed to the ceiling. The string passes under a movable pulley and then over a fixed pulley; its free end moves downward with acceleration a. The acceleration of the movable pulley is:
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Question

One end of a light inextensible string is fixed to the ceiling. The string passes under a movable pulley and then over a fixed pulley; its free end moves downward with acceleration a. The acceleration of the movable pulley is:

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G1 Physics · Accepted Answer

If x is the downward displacement of the free end and y that of the movable pulley, constant string length gives x + 2y = constant. Differentiating twice: a_x + 2a_y = 0. Therefore if the free end accelerates downward with a, the movable pulley accelerates upward with a/2.