Question 20: Oscillations

Question

A uniform disc of radius $R$ and mass $M$ is free to oscillate about the axis $A$ as shown in the figure. For small oscillations the time period is $\_\_\_\_$ .
(g is acceleration due to gravity)
![](https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1mnuintxg/837d1ca7-4973-45c4-b4d6-bad0dabe9c23/dfd9b040-35be-11f1-bedb-8be0f67d0034/file-6y3zli1mnuintxh.png?format=png)

Accepted Answer

Given,
$ x = a \sin \left(50t + \frac{\pi}{3}ight) $
This is a case of simple harmonic motion, where
$ \omega = 50 	ext{ rad/s} $
We need to find:
$t_1$: time when particle comes to rest, i.e. velocity $v=0$
$t_2$: time when acceleration is zero, i.e. $a=0$
First, velocity:
$ v = \frac{dx}{dt} = a \cdot 50 \cos\left(50t + \frac{\pi}{3}ight) $
For the particle to come to rest,
$ v=0 $
So,
$ \cos\left(50t + \frac{\pi}{3}ight)=0 $
Now, $\cos 	heta = 0$ when
$ 	heta = \frac{\pi}{2}, \frac{3\pi}{2}, \dots $
Taking the first time,
$ 50t + \frac{\pi}{3} = \frac{\pi}{2} $
$ 50t = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} $
$ t_1 = \frac{\pi}{300}	ext{ s} $
Now, acceleration in SHM is
$ \alpha = \frac{d^2x}{dt^2} = -\omega^2 x $
So acceleration becomes zero when displacement becomes zero:
$ x=0 $
That means,
$ a\sin\left(50t + \frac{\pi}{3}ight)=0 $
So,
$ \sin\left(50t + \frac{\pi}{3}ight)=0 $
Now, $\sin	heta=0$ when
$ 	heta = 0, \pi, 2\pi, \dots $
We take the first positive time. So,
$ 50t + \frac{\pi}{3} = \pi $
$ 50t = \pi - \frac{\pi}{3} = \frac{2\pi}{3} $
$ t_2 = \frac{2\pi}{150} = \frac{\pi}{75}	ext{ s} $
Hence,
$ t_1 = \frac{\pi}{300}	ext{ s}, \quad t_2 = \frac{\pi}{75}	ext{ s} $
So the correct option is:
$ \boxed{	ext{Option A}} $

Answer

$ 2 \pi \sqrt{\frac{5 R}{4 g}} $

Answer

$ 2 \pi \sqrt{\frac{2 R}{3 g}} $

Answer

$ 2 \pi \sqrt{\frac{3 R}{2 g}} $

Answer

$ 2 \pi \sqrt{\frac{3 R}{g}} $

Oscillations

A uniform disc of radius R and mass M is free to oscillate about the axis A as shown in the figure. For small oscillations the time period is \_\_\_\_ . (g is acceleration due to gravity)

Choose the correct answer: