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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


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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