Try this problem, useful for Physics Olympiad based on Maximum Height of Object over a Pulley.

The Problem: Maximum Height of Object over a Pulley

Two objects with masses 5Kg and 2Kg hang 0.6m above the floor from the ends of a cord 6m long passing over a frictionless pulley. Both objects start from rest. Find the maximum height reached by the 2.00Kg object.

Diagram of the pulley

Discussion:

Set up : After the (5Kg) object reaches the floor, the (2Kg) object is in free fall, with downward acceleration (g).

Execution: The (2Kg) will accelerate upward at$$ \frac{5-2}{5+2}g=3g/7$$ and the (5Kg) object will accelerate downward at (3g/7).

Let the initial height above the ground be (h_0).

When the large object hits the ground, the small object will be at a height (2h_0) and moving upward with a speed given by $$ v_0^2=2ah_0=6gh_0/7$$. The small object will rise to a distance (v_0^2/2g=3h_0/g) and so the maximum height reached will be $$ 2h_0+3h_0/7=17h_0/7=1.46m$$ above the floor, which is (0.860m) above its initial height.

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