Ratio of Displacements for Block Connected to Springs with Unattached Supports

To find the ratio of the displacements $\frac{y}{x}$, we analyze the mechanical energy of the system at the two extreme positions using the Principle of Conservation of Mechanical Energy.

1. Initial State (Displacement $x$ towards Wall 1):

• When the block $B$ is displaced to the right by a distance $x$ towards Wall 1:
  • Support $M_1$ rests against Wall 1, preventing further movement to the right. Thus, spring $S_1$ (with spring constant $k_1 = k$) is compressed by a distance $x$.
  • Support $M_2$ is not attached to Wall 2, so as the block moves right, $M_2$ simply slides to the right. Spring $S_2$ (with spring constant $k_2 = 4k$) remains at its relaxed (natural) length.
• Since the block is released from rest, the initial mechanical energy of the system is entirely potential energy stored in spring $S_1$:
  $E_i = \frac{1}{2} k x^2$

2. Motion towards Wall 2 (Displacement $y$ towards Wall 2):

• Upon release, spring $S_1$ pushes the block back towards the equilibrium position ($x = 0$). At $x = 0$, spring $S_1$ reaches its natural length, and support $M_2$ makes contact with Wall 2.
• As the block continues moving to the left beyond the equilibrium position by a maximum distance $y$:
  • Support $M_2$ is pressed against Wall 2 and cannot move further left, causing spring $S_2$ to be compressed by distance $y$.
  • Support $M_1$ is not attached to Wall 1, so it detaches from Wall 1 and moves along with the block to the left. As a result, spring $S_1$ remains at its natural length and stores zero potential energy.
• At the maximum displacement $y$, the block momentarily comes to rest. The final mechanical energy of the system is entirely potential energy stored in spring $S_2$:
  $E_f = \frac{1}{2} (4k) y^2$

3. Conservation of Energy:

Since all surfaces are frictionless and the springs/supports are massless, the total mechanical energy is conserved ($E_i = E_f$):

$\frac{1}{2} k x^2 = \frac{1}{2} (4k) y^2$

$k x^2 = 4k y^2$

$\frac{y^2}{x^2} = \frac{1}{4}$

Taking the square root on both sides:

$\frac{y}{x} = \frac{1}{2}$

Thus, the correct option is (C).