A block of mass m is suddenly released from the top of a spring of stiffness constant k.

a. Find the acceleration of block just after release.
b. What is the compression in the spring at equilibrium.
c. What is the maximum compression in the spring.
d. Find the acceleration of block at maximum compression in the spring.

Explanation

a. Just after release the spring is relaxed. Net force on the block is only its weight. Hence, acceleration of the block is g downward.
b. At equilibrium, net force on the block should be zero. Let `x_0` be the compression in the spring at equilibrium.

Applying Newton's second law,
`mg=kx_0` or `x_0=(mg)/(k)`
c. At maximum compression, the velocity of the block should be zero.
Let `x_(max)` be the maximum compression in the spring.
If we take block and spring as system, no external force and non-conservative force is doing work.

Applying mechanical energy conservation `DeltaK+DeltaU=0`
As the block is released from rest net charge and finally comes to rest. Hence, net change in its kinetic energy will be zero.
Net change in potential energy of the sytem
`DeltaU=DeltaU_(gr)+DeltaU_(sp)=(-mgx_(max))+(1/2kx_(max)^2)=0`
`implies0+(-mgx_(max)+1/2kx_(max)^2)=0` or `x_(max)=(2mg)/(k)`
d. At maximum compression, writing equation of motion of block
`kx_(max)-mg=ma` where `x_(max)=(2mg)/(k)`
Hence, acceleration of the block is g upward.