A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $π : 22$ then, the value of its angular speed will be _________ rad/s.

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answer is 4.

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Detailed Solution

$\frac{L}{E} = \frac{Iω}{\frac{1}{2} {mv}^{2} + \frac{1}{2} {Iω}^{2}}$

$= \frac{\frac{2}{5} {mR}^{2} ω}{\frac{1}{2} {mv}^{2} + \frac{1}{2} . \frac{2}{5} {mR}^{2} ω^{2}}$

$= \frac{\frac{2}{5} {mR}^{2} ω}{\frac{1}{2} {mR}^{2} ω^{2} + \frac{1}{5} {mR}^{2} ω^{2}}$
$= \frac{\frac{2}{5} {mR}^{2} ω}{\frac{7}{10} {mR}^{2} ω^{2}} = \frac{4}{7 ω}$

$Given \frac{L}{E} = \frac{π}{22} = \frac{4}{7 ω} = ω = 4$

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