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}} $

b
(b) ${E_1} = \frac{1}{2}K{x^2}$

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

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

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

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

$\Rightarrow x + y = \sqrt {\frac{{2E}}{K}} $

$ \Rightarrow \sqrt {\frac{{2{E_1}}}{K}} + \sqrt {\frac{{2{E_2}}}{K}} = \sqrt {\frac{{2E}}{K}} $

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

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