In an ideal pulley particle system, mass `m_2` is connected with a vertical spring of stiffness `k`. If mass `m_2` is released from rest, when the spring is underformed, find the maximum compression of the spring.

Answer

`(2(m_2-m_1)g)/(k)`

Explanation

Since work done by the constrain forces, tension and normal reaction is zero. The mechanical energy of the system will be conserve
`DeltaME=0`
or `DeltaK+DeltaU=0`

Let x be the maximum compression of the spring.
Then, `DeltaU_(sp)=k/2x^2` (i)
As `m_1` descends by a distance x,
`DeltaU_(gr_1)=-m_2gx`(ii)
As `m_2` ascends by a distance x, ltbrrgt `DeltaU_(gr_2)=m_2gx` (iii)
Since, both `m_1` and `m_2` comes to rest simultaneously,
`DeltaK=0` (iv)
Conservation of energy: `DeltaK+DeltaU=0`
`[DeltaU_(sp)+DeltaU_(gr_1)+DeltaU_(gr_2)]+DeltaK=0`
`(1/2kx^2+m_1gx-m_2gx)+0=0`
`impliesx=(2(m_2-m_1)g)/(k)`