Time period of simple pendulum is T.the time taken to complete 58 oscillations starting from mean position is αβT. The value of α+β is ______

58 Oscillation=12 oscillation +18 oscillation T2 for 12 oscillationFor 18 oscillation x=0 to x=A2, ⇒t=T12∴ Required time =T2+T12=7T12

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Time period of simple pendulum is T.the time taken to complete $\frac{5}{8}$ oscillations starting from mean position is $\frac{α}{β} T$ . The value of $α + β$ is ______

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answer is 19.

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Detailed Solution

$\frac{5}{8}$ Oscillation $= \frac{1}{2}$ oscillation $+ \frac{1}{8}$ oscillation
$\frac{T}{2}$ for $\frac{1}{2}$ oscillation
For $\frac{1}{8}$ oscillation $x = 0 to x = \frac{A}{2}$ ,
$\Rightarrow t = \frac{T}{12}$
$∴$ Required time $= \frac{T}{2} + \frac{T}{12} = \frac{7 T}{12}$