First slide

Simple hormonic motion

Question

A body is executing a simple harmonic motion such that its potential energy is $U_{1}$ at x and $U_{2}$ at y. When the displacement is x+y , the potential energy will be

Moderate

A

$U_{1} + U_{2}$
B

$\sqrt{{U_{1}}^{2} + {U_{2}}^{2}}$
C

$U_{1} + U_{2} + 2 \sqrt{U_{1} U_{2}}$
D

$\sqrt{U_{1} U_{2}}$

Solution

At displacement x, $U_{1} = \frac{1}{2} {mω}^{2} x^{2}$
$\Rightarrow x = \sqrt{\frac{2 U_{1}}{{mω}^{2}}}$
Similarly, $y = \sqrt{\frac{2 U_{2}}{{mω}^{2}}}$
When displacement is (x+y), potential energy is
$U = \frac{1}{2} {mω}^{2} {(x + y)}^{2} = \frac{1}{2} {mω}^{2} (x^{2} + y^{2} + 2 \sqrt{xy})$
= $\frac{1}{2} {mω}^{2} x^{2} + \frac{1}{2} {mω}^{2} y^{2} + {mω}^{2} \sqrt{xy}$
= $U_{1} + U_{2} + {mω}^{2} \sqrt{\frac{2 U_{1}}{{mω}^{2}} . \frac{2 U_{2}}{{mω}^{2}}}$
= $U_{1} + U_{2} + \sqrt[2]{U_{1} U_{2}}$

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