First slide

Simple hormonic motion

Question

A simple pendulum has time period $T_{1}$ . The point of suspension is now moved upward according to equation y = k $t^{2}$ where k = 1 $m / \sec^{2}$ . If new time period is $T_{2}$ , then ratio $\frac{T_{1}^{2}}{T_{2}^{2}}$ will be

Moderate

A

$\frac{2}{3}$
B

$\frac{5}{6}$
C

$\frac{6}{5}$
D

$\frac{3}{2}$

Solution

$y = {Kt}^{2} \Rightarrow \frac{d^{2} y}{{dt}^{2}} = a_{y} = 2 K = 2 \times 1 = 2 m / s^{2}$
$(∵ K = 1 m / s^{2})$
Now, $T_{1} = 2 π \sqrt{\frac{l}{g}} a n d T_{2} = 2 π \sqrt{\frac{l}{(g + a_{y})}}$
Dividing, $\frac{T_{1}}{T_{2}} = \sqrt{\frac{g + a_{y}}{g}} = \sqrt{\frac{6}{5}} \Rightarrow \frac{T_{1}^{2}}{T_{2}^{2}} = \frac{6}{5}$

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