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Starting from the rest, at the same time, a ring, a coin and a solid ball of same mass roll down an incline without slipping .The ratio of their translational kinetic energies at the bottom will be
$1 : 1 : 1$
$10 : 5 : 4$
$21 : 28 : 30$
None
Solution
Using conservation of energy.
$m g h=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}$
$m g h=\frac{1}{2} m v^{2}+\frac{1}{2} m k^{2} \omega^{2}$ where $k:$ radius of gyration
$m g h=\frac{\overline{1}}{2} m v^{2}+\frac{\overline{1}}{2} m v^{2} \frac{k^{2}}{r^{2}}$
$K E=\frac{P E}{1+k^{2} / r^{2}}$
$\Longrightarrow K E_{\text {ring}}: K E_{\text {coin}}: K E_{\text {solid} b a l l}=\frac{1}{1+1}: \frac{1}{1+1 / 2}: \frac{1}{1+2 / 5}$
$K E_{\text {ring}}: K E_{\text {coin}}: K E_{\text {solid} b a l l}=21: 28: 30$
