A body is executing Simple Harmonic Motion. At a displacement x i… - Vidyadip

MCQ

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
$\sqrt{E}=\sqrt{E_1}-\sqrt{E_2}$
✓
$\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$
C
$E=E_1+E_2$
D
$E=E_1+E_2$

✓

Answer

Correct option: B.

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

$ E_1=\frac{1}{2} K x^2 \Rightarrow x=\sqrt{\frac{2 E_1}{K}}, E_2=\frac{1}{2} K y^2 \Rightarrow y=\sqrt{\frac{2 E_2}{K}} $
$ \text { and } E=\frac{1}{2} K(x+y)^2 \Rightarrow x+y=\sqrt{\frac{2 E}{K}} $
$ \Rightarrow \sqrt{\frac{2 E_1}{K}}+\sqrt{\frac{2 E_2}{K}}=\sqrt{\frac{2 E}{K}} \Rightarrow \sqrt{E_1}+\sqrt{E_2}=\sqrt{E}$

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