Question

A body executing SHM at a displacement 'x' its PE is $E_1$, at a displacement 'Y' its PE is $E_2$ The P.E at a displacement $(x+y)$ is

• A. $\sqrt{E}=\sqrt{E_1}-\sqrt{E_2}$
• B. $\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$
• C. $E=E_1+E_2$
• D. $E=E_1-E_2$

Answer 1

The correct option is B $\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$

In $SHM$

$PE=1/0K{x}^2$, where $K=m{w}^2$

For displacement $x$

$PE=E_1=\frac{1}{2}K{x}^2\Rightarrow x=\sqrt{\frac{2E_1}{k}}---\left(i\right)$

For displacement $y$

$PE=E_2=\frac{1}{2}K{y}^2=y\sqrt{\frac{2E_2}{K}}$

For displacement $x+y$

$PE=\frac{1}{2}K[x+y{]}^2$

$E=\frac{1}{2}K{[\sqrt{\frac{2E_1}{K}}+\sqrt{\frac{2E_2}{K}}]}^2$

$\Rightarrow E=\left(\frac{2}{2}\right)\frac{K}{K}[\sqrt{E_1}+\sqrt{E_2}{]}^2$

$\Rightarrow E=[\sqrt{E_1}+\sqrt{E_2}{]}^2$

$\Rightarrow \sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$

Option $B$ is correct