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
Diffcult
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(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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