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A smooth block of mass m is released fro...

A smooth block of mass m is released from rest from a height h. It slides and compresses the spring of stiffness k. Find the maximum compression of the spring .
Taking block spring a system. Now external force is doing work on system and no non-conservative force is present. The mechanincal energy should be conserved.
`DeltaK+DeltaU=0`

Text Solution

Verified by Experts

Let x=maximum compression of the spring.

Here `DeltaU=DeltaU_(sp)+DeltaU_(gr)`
`DeltaU_(sp)=k/2x^2`
because the spring is deformed from `x=0` to `x=x` and `DeltaU_(gr)=-mgh` because the block falls down through a vertical distance h. Hence,
`DeltaU=k/2x^2-mgh`
Since, the block will come to rest at the time of maximum compression `DeltaK=0`. Substituting `DeltaU` and `DeltaK` in the equation
`DeltaU+DeltaK=0`, we have `k/2x^2-mgh=0`
Then `x=sqrt((2mgh)/(k))`
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