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Potential Energy 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

(Assume that potential energy at mean position is zero)

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 $E_{1} + E_{2} + 2 \sqrt{E_{1} E_{2}}$
Potential energy of a body executing SHM for a displacement $x$ is given by
$E_{1} = \frac{1}{2} k x^{2} . . . . . (1)$
Potential energy of a body executing SHM for a displacement $y$ is given by
$E_{2} = \frac{1}{2} k y^{2} . . . . . . . (2)$
Potential energy of a body executing SHM for a displacement $(x + y)$ is given by
$E = \frac{1}{2} k (x + y)^{2}$
$\Rightarrow E = \frac{1}{2} k (x^{2} + y^{2} + 2 x y)$
From $(1)$ and $(2)$ we get,
$E = E_{1} + E_{2} + k \sqrt{\frac{2 E_{1}}{k}} \sqrt{\frac{2 E_{2}}{k}}$
$\Rightarrow E = E_{1} + E_{2} + 2 \sqrt{E_{1} E_{2}}$
Thus, option (c) is the correct answer.

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Potential Energy Formulae

Standard XII Physics

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