A block $(B)$ is attached to two unstretched springs $S_1$ and $S_2$ with spring constants $k$ and $4k$, respectively (see fig. I). The other ends are attached to identical supports $M_1$ and $M_2$ not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block $B$ is displaced towards wall 1 by a small distance $x$ (figure II) and released. The block returns and moves a maximum distance $y$ towards wall 2. Displacements $x$ and $y$ are measured with respect to the equilibrium position of the block $B$. The ratio $y/x$ is -

Step 1: Analyze the initial displacement towards Wall 1.

When the block $B$ is displaced towards Wall 1 (to the right) by a distance $x$, support $M_1$ is pressed against Wall 1. Thus, spring $S_1$ (with spring constant $k$) is compressed by $x$.

Since support $M_2$ is massless and free, it slides to the right by $x$ along with the block, keeping spring $S_2$ relaxed.

The potential energy stored in the system at this initial position is:

$$U_i = \frac{1}{2} k x^2$$

Step 2: Analyze the motion past the equilibrium position towards Wall 2.

Upon release, the block moves back to the equilibrium position, converting the stored potential energy into kinetic energy.

As the block continues past the equilibrium position towards Wall 2 (to the left) by a maximum distance $y$, support $M_2$ is pinned against Wall 2. This causes spring $S_2$ (with spring constant $4k$) to compress by $y$.

Meanwhile, the massless support $M_1$ is pulled away from Wall 1 and moves freely with the block, keeping spring $S_1$ relaxed.

The potential energy stored in the system at the maximum displacement $y$ is:

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

Step 3: Apply conservation of mechanical energy.

Since there is no friction, the total mechanical energy of the system is conserved.

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

Simplify the equation to find the ratio of the displacements.

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

Answer: The ratio $y/x$ is $\dfrac{1}{2}$ (or $0.5$).