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A particle executing simple harmonic motion, represented by $x=a \sin \omega t$ takes $t_{1}$ time in going from $x=0$ to $x=\frac{a}{2}$ and the time $t^{2}$ in going from $x=\frac{a}{2}$ to $x=a$. The ration $t_{1}$:$t_{2}$ would be..?

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$\frac{a}{2} = a \sin \omega t_{1}$

$\sin \omega t_{1} = \frac{1}{2} = \sin 30^{\circ}$

$\Rightarrow \omega t_{1} = 30^{\circ}$

$a = a \sin \omega (t_1+t_2) \sin \omega (t_1+t_2) = \sin 90^{\circ}$

$\Rightarrow \omega (t_1+t_2) = 90^{\circ}$

$\therefore \frac{\omega t_1}{\omega(t_1+t_2)} = \frac{30^{\circ}}{90^{\circ}} = \frac{1}{3} $

$\Rightarrow \frac{t_1}{t_2} = \frac{1}{2}$
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