Question

A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4k, respectively (see figure I).The other ends are attached to identical supports M1 and M2 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 $\frac{y}{x}$ is

• A. $4$
• B. $2$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{1}{2}$

At first the block B moves towards the wall 1 and due to this the spring $S_1$ is compressed by $xunits.$ After that, the block moves towards the wall 2 which compresses the spring $S_2$ by $yunits.$

Velocity of the block is zero in both cases whereas springs $S_1$ and $S_2$ have respective potential energies as $P.E_C$ and $P.E_D$.

Applying energy conservation at points $C$ and $D:$

$K.E_C+P.E_C=K.E_D+P.E_D$

$0+\frac{1}{2}{k}_{s_1}{x}^2=0+\frac{1}{2}{k}_{S_2}{y}^2$

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

$⟹\frac{y}{x}=\frac{1}{2}$