As shown in the figure, a bob of mass m


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 $\mathrm{r}$ and mass $\mathrm{m}$. When released from rest the bob starts falling vertically. When it has covered a distance of $\mathrm{h}$, the angular speed of the wheel will be:

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

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

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

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

Correct Option: 1


(1) When the bob covered a distance ' $h$ '

Using $m g h=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}$

$=\frac{1}{2} \mathrm{~m}(\omega \mathrm{r})^{2}+\frac{1}{2} \times \frac{m r^{2}}{2} \times \omega^{2}(\because v=\omega r$ no slipping $)$

$\Rightarrow \quad m g h=\frac{3}{4} m \omega^{2} r^{2}$

$\Rightarrow \omega=\sqrt{\frac{4 g h}{3 r^{2}}}=\frac{1}{r} \sqrt{\frac{4 g h}{3}}$

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