A particle performs $SHM$ about $x = 0$ such that at $t = 0$ it is at $x = 0$ and moving towards positive extreme. The time taken by it to go from $x = 0$ to $x = \frac{A}{2}$ is ..... times the time taken to go from $x = \frac{A}{2}$ to $A$. The most suitable option for the blank space is
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Time for $x=0$ to $x=\frac{A}{2}$ is
$\frac{\mathrm{A}}{2}=\mathrm{A} \sin \omega \mathrm{t}_{1} \quad \Rightarrow \omega \mathrm{t}_{1}=\frac{\pi}{6} \Rightarrow \mathrm{t}_{1}=\frac{\pi}{6 \omega}$
$\Rightarrow t_{1}=\frac{T}{12}$
and from $x=\frac{A}{2}$ to $x=A$ is
$\mathrm{t}_{2}=\frac{\mathrm{T}}{4}-\frac{\mathrm{T}}{12}=\frac{\mathrm{T}}{6} \Rightarrow \frac{\mathrm{t}_{1}}{\mathrm{t}_{2}}=\frac{1}{2}$
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