how do i this kind of ques ?i think that conservation of energy is required in this ques. but I am not able to apply it.please help me.

Let relaxed length of the spring is L. Let coordinates of block is x₁, coordinates of point P is x₂ and elongation of the spring is x at certain time t. Then we can write

Differentiating twice wrt t, we can write

                                                                                    (1)

Given that acceleration of P is a₂ (= 4 m/s²), and acceleration of the block is

We can write the above equation as

                                                                          (2)

Solution of this differential equation gives

, where

Acceleration a₁ of the block is given by

                                                 (3)

To find maximum and minimum values of a₁,

From this we can easily see that a₁ will have minimum value when ωt = 0, 2π, etc and will be maximum at ωt = π, 3π, etc..

Hence from equation 3, minimum value of a₁ = 0, whereas maximum value of a₁ = 8 m/s²

Alternatively, let us try to solve this problem from the reference frame of point P. Viewed from this frame, there will be a pseudo force -ma_p acting on the mass m, where a_p is the accleration of P. You can see that since in this frame point P is fixed, the mass m will have Simple Harmonic Oscillations as it is attached to the spring.

I am assuming that you are aware of Simple Harmonic Motion (SHM) concepts, then you will know that maximum acceleration of the mass will happen at extreme position (maximum extension).

At the maximum extension position, the mass will come to instantaneous rest, and work done by the pseudo force will be equal to the potential energy stored in the spring.

So, if x is the maximum extension, then we can write and at the maximum extension, acceleration of the mass is given by , as a_p = 4 m/s², given.

Hope this helps.