use formula , of , conservation of energym*g*(h+x) = kx²/2where x is compression2 ( mgh + mgx ) = kx²k x² – 2mgx – 2mgh = 0and solve this quadratic equation for getting value of x , which is compression

**Problem Statement:**
A block of mass m is released from rest from a height h above the light pan attached to a vertical spring of force constant k mounted on the floor. We need to find the maximum compression in the spring.

**Solution:**
To find the maximum compression in the spring, we need to analyze the energy transformation during the motion of the block.

1. **Initial State:**
- The block is at rest at a height h above the light pan.
- The spring is in its equilibrium position.
- The total mechanical energy in the system is given by the potential energy of the block due to its height above the pan.
- The potential energy is given by PE = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height.

2. **Release of the Block:**
- When the block is released, it starts falling towards the pan due to the force of gravity.
- As the block falls, its potential energy is converted into kinetic energy.
- At any point during the fall, the mechanical energy is given by the sum of the kinetic energy and the potential energy.
- The kinetic energy of the block is given by KE = (1/2)mv^2, where v is the velocity of the block.

3. **Impact with the Spring:**
- When the block reaches the pan, it compresses the spring.
- As the block compresses the spring, its kinetic energy is converted into potential energy stored in the spring.
- The potential energy stored in the spring is given by PE_spring = (1/2)kx^2, where k is the force constant of the spring and x is the compression of the spring.
- The compression of the spring can be calculated using Hooke's Law: F = kx, where F is the force exerted by the spring.

4. **Maximum Compression:**
- The maximum compression in the spring occurs when all the kinetic energy of the block is converted into potential energy stored in the spring.
- At this point, the velocity of the block becomes zero, and all the mechanical energy is stored in the spring.
- Equating the kinetic energy at the moment of impact to the potential energy stored in the spring, we get:
(1/2)mv^2 = (1/2)kx^2
- Solving for x, we get:
x = sqrt((mv^2)/k)

Hence, the maximum compression in the spring is given by x = sqrt((mv^2)/k), where m is the mass of the block, v is the velocity of the block at the moment of impact, and k is the force constant of the spring.