Question

A block $\left(B\right)$ is attached to two unstretched springs $S_1$ and $S_2$ with spring constant $2k$ and $4k$ respectively. The other ends are attached to two supports $m_1$ and $m_2$ not attached to the walls. The springs and supports have negligible mass and the surface is frictionless. The block $B$ is displaced towards wall $1$ by a small distance $x$ and released. The block returns and moves a maximum distance $y$ towards wall $2$. Displacement $x$ and $y$ is measured with respect to the equilibrium position of the block $B$. The ratio of $\frac{y}{x}$ is

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

Answer 1

The correct option is D $\frac{1}{\sqrt{2}}$

Given that,
Spring constant of spring $S_1=2k$
Spring constant of spring $S_2=4k$
Initial compression of spring $S_1=x$
Energy stored in spring $S_1=\frac{1}{2}\times 2k{x}^2$
$=k{x}^2\dots \dots \left(i\right)$
The block will return from wall $1$ and moves toward wall $2$ and spring $S_2$ will be compressed.


Energy stored in $S_2=\frac{1}{2}4k{y}^2=2k{y}^2$
By energy conservation,
Energy stored in spring $S_1$ initially $=$ Energy stored in $S_2$ finally
$\Rightarrow k{x}^2=2k{y}^2$
$\Rightarrow \frac{x^2}{y^2}=\frac{2}{1}$
$\Rightarrow \frac{y}{x}=\frac{1}{\sqrt{2}}$