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Which of the following (if mass and radius are assumed to be same) have maximum percentage of total $K.E.$ in rotational form while pure rolling?
Disc
Sphere
Ring
Hollow sphere
Solution
(c)
$K _{\text {Rother }}= K _{\text {Trenstation }}+ K _{\text {rotaton }}$
$=\frac{1}{2} m v^2+\frac{1}{2} l \omega^2$
$=\frac{1}{2} m R^2 \omega^2+\frac{1}{2}\left|\omega^2=\frac{1}{2}\right| \omega^2\left(1+\frac{m R^2}{1}\right)$
$\Rightarrow$ ratio $=\frac{K_{\text {waton }}}{K_{\text {total }}}=\frac{\frac{1}{2} I ^2}{\frac{1}{2} I_{\omega^2}\left(1+\frac{m R^2}{1}\right)}$
$=\frac{1}{1+\frac{m R^2}{1}}$
For disc,
$\text { Ratio }=\frac{1}{1+\frac{m R^2}{\left(\frac{m R^2}{2}\right)}}=\frac{1}{3}$
Solid sphere
$\text { Ratio }=\frac{1}{1+\frac{m R^2}{\frac{2}{5} m R^2}}=\frac{2}{7}$
Ring
$\text { Ratio }=\frac{1}{1+\frac{m R^2}{m R^2}}=\frac{1}{2}$
Hollow sphere ratio $=\frac{1}{1+\frac{m R^2}{\frac{2}{3} m R^2}}=\frac{2}{5}$
Hence $ring$ correct option.
