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A solid uniform sphere rotating about its axis (with rotational kinetic energy $ω_{0}$ ) is placed on a rough horizontal plane without any translational push. Friction coefficient p is not same everywhere on the plane and it may differ even at each point. After sometime the sphere begins pure rolling with total kinetic energy E. Then:

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a

$ω$ cannot be expressed in terms $ω_{0}$

b

$ω = 2 ω_{0} / 5$

c

$ω = 2 ω_{0} / 7$

d

$ω = 5 ω_{0} / 7$

answer is B.

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Detailed Solution

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L_i = L_f
$\begin{array}{l} \Rightarrow (\frac{2}{5} {MR}^{2}) ω_{0} = MVR + (\frac{2}{5} {MR}^{2}) ω \\ = ω = \frac{2}{7} ω_{0} \\ {TKE}_{i} = \frac{1}{2} (\frac{2}{5} {MR}^{2}) ω_{0}^{2}; {TKE}_{f} = \frac{1}{2} {mv}^{2} + \frac{1}{2} {Iω}^{2} = \frac{7}{10} {Mω}^{2} R^{2} \\ = \frac{2}{35} {MR}^{2} ω_{0}^{2} \end{array}$

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