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Spring Block's Time Period

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Question

The acceleration-displacement (a-x) graph of a particle executing simple harmonic motion is shown in the figure. Find the frequency of its oscillation.

A

\frac{1}{2 π} \sqrt{\frac{2 β}{α}}

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B

\frac{1}{2 π} \sqrt{\frac{β}{2 α}}

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C

\frac{1}{2 π} \sqrt{\frac{β}{α}}

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D

\frac{1}{2 π} \sqrt{\frac{β}{α - β}}

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Solution

The correct option is B
$\frac{1}{2 π} \sqrt{\frac{β}{α}}$

Answer: 1
For Simple Harmonic Motion,
$a (t) = - ω^{2} x (t)$
Differentiate with respect to x.
$\frac{d a}{d x} = - ω^{2} \frac{d x}{d x} = - ω^{2}$
From graph given the slope is $- \frac{β}{α}$
Therefore, $\frac{d a}{d x} = - \frac{β}{α}$
$ω = \sqrt{\frac{β}{α}}$
Finally, frequency of oscillation, $f = \frac{ω}{2 π} = \frac{1}{2 π} \sqrt{\frac{β}{α}}$

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