Question

A body is executing SHM, At a displacement x, its potential energy is $E_1$ and at a displacement y, its potential energy is $E_2$. Find its potential energy E at displacement ( x + y)

• A. $\sqrt{E}={\sqrt{E}}_1-{\sqrt{E}}_2$
• B. $\sqrt{E}={\sqrt{E}}_1+{\sqrt{E}}_2$
• C. none of these
• D. all of these

Answer 1

The correct option is B $\sqrt{E}={\sqrt{E}}_1+{\sqrt{E}}_2$

Potential energy at $x=E_1=\frac{1}{2}K{x}^2$

$x=\sqrt{\frac{2E_1}{K}}$

Potential energy at $y=E_2=\frac{1}{2}K{y}^2$

$y=\sqrt{\frac{2E_2}{K}}$

Potential energy at $x+y=E=\frac{1}{2}K(x+y{)}^2$

$E=\frac{1}{2}K(\sqrt{\frac{2E_1}{K}}+\sqrt{\frac{2E_2}{K}})=(E_1+E_2{)}^2$

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