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SHM Formulae

A body is exe...

Question

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 a displacement $(x + y)$ is

E_{1} + E_{2}

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\sqrt{E_{1}^{2} + E_{2}^{2}}

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E_{1} + E_{2} + 2 \sqrt{E_{1} E_{2}}

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\sqrt{E_{1} E_{2}}

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Solution

The correct option is C $E_{1} + E_{2} + 2 \sqrt{E_{1} E_{2}}$
At displacement x
$E_{1} = \frac{1}{2} m ω^{2} x^{2}$
or $\sqrt{E_{1}} = x \sqrt{\frac{1}{2} m ω^{2}}$
At displacement y
$E_{2} = \frac{1}{2} m ω^{2} y^{2}$
or $\sqrt{E_{2}} = y \sqrt{\frac{1}{2} m ω^{2}}$
At displacement $(x + y)$
$E = \frac{1}{2} m ω^{2} (x + y)^{2}$
$\sqrt{E} = (x + y) \sqrt{\frac{1}{2} m ω^{2}}$
Hence,
$\sqrt{E} = \sqrt{E_{1}} + \sqrt{E_{2}}$
$E = E_{1} + E_{2} + 2 \sqrt{E_{1} E_{2}}$

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