A block of mass `m` held touching the upper end of a light spring of force constant K as shown in figure. Find the maximum potential energy stored in the spring if the block is released suddenly on the spring.

Explanation

Since the system is conservative the total energy E of the system remains constant
E=constant.
`implies DeltaE=0`
`implies DeltaU+DeltaKE=0`
`implies(DeltaU)_(spri ng)+(DeltaU)_(gr)+DeltaKE=0` (1)

where `(DeltaU)_(spri ng)`=change in potential energy of the spring `=(1//2)`
`Kx^2` since the spring is compressed through a distance x.
`DeltaU_(gr)=` change in gravitational potential energy of the block as the block falls through x, `DeltaU=-mgx`.
As the block is released from rest and comes to rest after compressing the spring through a length x the change in KE is zero `impliesDeltaKE=0`.
Putting the `(DeltaU)_(spri ng)=1/2kx^2`, `(DeltaU)_gr=-mgh`, and `DeltaKE=0`, we get, `1/2kx^2-mgx=0=x=(2mg)/(k)`
Thus, the maximum potential energy stored in the spring
`=1/2k((2mg)/(k))^2=(2m^2g^2)/(k)`