Question

A body is executing Simple Harmonic Motion. At a displacement x its potential energy is $E_{1}$ and at a displacement y its potential energy is$E_{2}$. The potential energy E at 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

Answer: (B)

$\sqrt{E} = \sqrt{E_{1}} + \sqrt{E_{2}}$

Answer 2

$E_{1} = \frac{1}{2}Kx^{2} \Rightarrow x = \sqrt{\frac{2E_{1}}{K}}$, $E_{2} = \frac{1}{2}Ky^{2} \Rightarrow y = \sqrt{\frac{2E_{2}}{K}}$

and $E = \frac{1}{2}K(x + y)^{2} \Rightarrow x + y = \sqrt{\frac{2E}{K}}$

$\Rightarrow \sqrt{\frac{2E_{1}}{K}} + \sqrt{\frac{2E_{2}}{K}} = \sqrt{\frac{2E}{K}} \Rightarrow \sqrt{E_{1}} + \sqrt{E_{2}} = \sqrt{E}$