Q. 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) $E = E_{1} + E_{2}$
Explanation
(b) $E_{1} = \frac{1}{2}Kx^{2} \Rightarrow x = \sqrt{\frac{2E_{1}}{K}}$
$E_{2} = \frac{1}{2}Ky^{2} \Rightarrow y = \sqrt{\frac{2E_{2}}{K}}$
and$\ E = \frac{1}{2}K(x + y)^{2} \Rightarrow x + y = \sqrt{\frac{2E}{K}}$
$\Rightarrow \sqrt{\frac{2E_{1}}{K}} + \sqrt{\frac{2E_{2}}{K}} = \sqrt{\frac{2E}{K}} \Rightarrow \sqrt{E_{1}} + \sqrt{E_{2}} = \sqrt{E}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.