Question

A block of mass $m$ is attached to two unstretched springs of spring constants $k_{1}$ and $k_{2}$ as shown in figure.The block is displaced towards right through a distance $x$ and is released. The speed of the block as it passes through the mean position is

• A. $x \sqrt{\frac{k_{1}}{m}}$
• B. $x \sqrt{\frac{k_{2}}{m}}$
• C. $x \sqrt{\frac{k_{1}+k_{2}}{m}}$
• D. $x \sqrt{\frac{k_{1}-k_{2}}{m}}$

Answer 1

Answer: (C)

$U_{i}=\frac{1}{2} k_{1} x^{2}+\frac{1}{2} k_{2} x^{2}$
$U_{f}=0 ; K_{i}=0 ; K_{f}=\frac{1}{2} m v^{2}$
Here, $ \Delta U+\Delta K=0$
$\left[0-\left(\frac{1}{2}\left(k_{1}+k_{2}\right) x^{2}\right)\right]+\left[\frac{1}{2} m v^{2}-0\right]=0$
$\Rightarrow \frac{1}{2} m v^{2}=\frac{1}{2}\left(k_{1}+k_{2}\right) x^{2}$
$\Rightarrow v=x \sqrt{\frac{\left(k_{1}+k_{2}\right)}{m}}$