First slide

Rotational motion

Question

As shown in the figure, a bob of mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. when released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be

Moderate

A

$\frac{1}{r} \sqrt{\frac{4 g h}{3}}$
B

$\frac{1}{r} \sqrt{\frac{2 g h}{3}}$
C

$r \sqrt{\frac{3}{2 g h}}$
D

$r \sqrt{\frac{3}{4 g h}}$

Solution

$\begin{array}{l} L o s s i n p o t e n t i a l e n e r g y i s g a i n i n t r a n s l a t i o n a l (b o b) a n d r o t a t i o n a l (d i s c) k i n e t i c e n e r g y . \\ \Rightarrow m g h = \frac{1}{2} m v^{2} + \frac{1}{2} l ω^{2} h e r e, v = ω r (n o s l i p p i n g) \\ \Rightarrow m g h = \frac{1}{2} m ω^{2} r^{2} + \frac{1}{2} \frac{m r^{2}}{2} ω^{2} \\ \Rightarrow m g h = \frac{3}{4} m ω^{2} r^{2} \\ \Rightarrow ω = \sqrt{\frac{4 g h}{3 r^{2}}} = \frac{1}{r} \sqrt{\frac{4 g h}{3}} \end{array}$

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