Two particles of same mass are performing simple harmonic oscillation along the $y -$ axis whose displacement time graphs are shown. If $E_{1}$ and $E_{2}$ are their energies of oscillation, then

E_{1} = 2 E_{2}

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E_{1} = 4 E_{2}

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E_{1} = E_{2}

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E_{2} = 4 E_{1}

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Solution

The correct option is C $E_{1} = E_{2}$
$E = \frac{1}{2} m ω^{2} A^{2} = \frac{1}{2} m (2 π v)^{2} A^{2}$
$= \frac{1}{2} m (4 π^{2} v^{2}) A^{2} = (2 m π^{2} v^{2} A^{2})$
For $1$ , Amplitude $= 2 A$ , frequency $= v$
$∴ E_{1} = 2 m π^{2} v^{2} (2 A)^{2} = 8 m π^{2} v^{2} A^{2}$
For $2$ , Amplitude $= A$ , frequency $= 2 v$
$∴ E_{2} = 2 m π^{2} (2 v)^{2} A^{2} = 8 m π^{2} v^{2} A^{2}$
i.e., $E_{1} = E_{2}$

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