Question

solve it with correct explaination........and plzzzz avoid spamming.........​

Answer 1

Answer:

The carpet has three kind of energies transational , rotational and potential .

When r = R

mass = m , rotational energy = 0 , transitional = 0

Potential energy =

$mg r$

When r = R/2

mass

$= \rho(\pi( { \frac{r}{2}) }^{2} l) = \frac{ \rho \: \pi {r}^{2}l }{4} = \frac{m}{4} \\ \{since \: m = \rho v = \rho\pi {r}^{2} l \}$

Rotational energy

$= \frac{1}{2} i { \omega}^{2} = \frac{1}{2} ( \frac{ \frac{m}{4} ( \frac{r}{2}) {}^{2} }{2} )( \frac{v}{ \frac{r}{2}}{) {}^{2} } = \frac{1}{16} m {v}^{2} \\ \\ i.e .\: \: \: v = wr$

Translational Energy

$= \frac{1}{2} \frac{m}{4}. {v}^{2} = \frac{m {v}^{2} }{8}$

Potential energy

$mgh = \frac{m}{4} .g. \frac{r}{2} = \frac{mgr}{8}$

$\pink{use \: energy \: conservation}$

$\green{rot. + ke + pe= const} \\ \\ 0 + 0 + mgr = \frac{m {v}^{2} }{16} + \frac{m {v}^{2} }{8} + \frac{mgr}{8} \\ \\ mgr(1 - \frac{1}{8} ) = m {v}^{2} ( \frac{1}{16} + \frac{1}{8} ) \\ \\ \frac{7mgr}{8} = \frac{3m {v}^{2} }{16} \\ \\ {v}^{2} = \frac{14gr}{3} \\ \\ v = \sqrt{ \frac{14gr}{ 3} }$