A particle executes SHM with amplitude $a$ and time period $T$ . The displacement of the particle from the mean position when its speed is half of maximum speed is $\frac{\sqrt{x} a}{2}$ . The value of $x$ is _______.

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Solution

For a particle executing SHM, $v = ω \sqrt{A^{2} - X^{2}}$

$\Rightarrow \frac{ω a}{2} = ω \sqrt{a^{2} - X^{2}}$

$\Rightarrow \frac{ω^{2} a^{2}}{4} = ω^{2} a^{2} - ω^{2} X^{2}$

$\Rightarrow X^{2} = \frac{3}{4} a^{2}$

$\Rightarrow X = \frac{\sqrt{3} a}{2} = \frac{\sqrt{x} a}{2}$

$\Rightarrow x = 3$

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