A block is released from height `h`, find the maximum compression of spring of spring constant `k`.

Explanation

Let the block compress the spring by an amount x before coming to rest for a moment as shown in figure.
Block is initially at rest and at the time of maximum compression it again comes to rest hence we can say that there is no change in kinetic energy of the sustem in this interval . Work is done on the system by gravity and spring force and both are conservative forces hence system is conservative and total mechnical energy of the system remains conserved .
Loss in gravitational poential energy of the block = gain in spring potential energy
` rArr " " mg( h+x) = 1/2 Kx^(2)`
`rArr " " 2mgh + 2mgx = Kx^(2)`
`rArr Kx^(2) - 2mgx - 2mgh = 0 `
`rArr " " x = (2mg+sqrt(4m^(2)g^(2)+8mghK))/(2K)`
Negative root is not acceptable , hence maximum compression can be written as follows :
` rArr " " x = (mg)/K [ 1+ sqrt(1+(2Kh)/(mg))] `