A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about the axis of rotation of the sphere to the total energy of the moving sphere is , then the value of its angular speed will be ________ rad/s. [20

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Solution

Q.
A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about the axis of rotation of the sphere to the total energy of the moving sphere is $π : 22$ , then the value of its angular speed will be ________ rad/s. [2023]

Ans.
(4)

$L = (I_{com}) (ω) and K = \frac{1}{2} (I_{com}) (ω^{2}) + \frac{1}{2} M V_{com}^{2}$

$L = \frac{2}{5} M R^{2} \frac{V_{com}}{R}$

$K = \frac{1}{2} (\frac{2}{5} M R^{2}) \frac{v_{com}^{2}}{R^{2}} + \frac{1}{2} M V_{com}^{2}$

$L = \frac{2 M R V_{com}}{5} K = \frac{7}{10} M V_{com}^{2}$

$Ratio \frac{L}{K} = \frac{4}{7} \frac{R}{v_{com}} = \frac{π}{22} \Rightarrow ω = \frac{4}{7} \times \frac{22}{22} \times 7 = 4$

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