In this question, we have a mass (m) attached to two springs, each with a spring constant (k). The system is in equilibrium, and we need to calculate the work required to lift the block up by a distance of mg/4k.


To solve this problem, we need to consider the forces acting on the mass when it is in equilibrium. At equilibrium, the net force acting on the mass is zero.


The force due to gravity acting on the mass is given by Fg = mg, where m is the mass and g is the acceleration due to gravity.


The force exerted by each spring can be calculated using Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position. Since the mass is in equilibrium, the forces exerted by the two springs must cancel each other out.


At equilibrium, the forces acting on the mass are balanced, so the net force is zero. This can be expressed as:

Fg + F1 + F2 = 0

where F1 and F2 are the forces exerted by the two springs.


To calculate the work required to lift the block by a distance of mg/4k, we need to consider the potential energy stored in the system. The work done is equal to the change in potential energy.

The potential energy stored in a spring is given by the formula U = 1/2 kx^2, where k is the spring constant and x is the displacement from the equilibrium position.

In this case, the displacement is mg/4k. Therefore, the potential energy stored in each spring is:

U = 1/2 k (mg/4k)^2 = m^2g^2/32k

Since there are two springs, the total potential energy stored in the system is:

U_total = 2 * (m^2g^2/32k) = m^2g^2/16k

The work done to lift the block is equal to the change in potential energy, which is:

Work = U_final - U_initial = (m^2g^2/16k) - 0 = m^2g^2/16k

Therefore, the work required to lift the block up by a distance of mg/4k is m^2g^2/16k.


The work required to lift the block up by a distance of mg/4k is m^2g^2/16k. This is calculated by considering the equilibrium condition of the system and the potential energy stored in the springs.