A sharp vertical impulse J is given to a uniform solid sphere of mass m and radius R placed on a rough horizontal surface. Choose the correct options. (Neglect impulse due to weight of the sphere and consider that sphere does not bounce up).

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Minimum value of coefficient of friction between sphere and surface to avoid slipping is $μ = \frac{4}{7} .$

Impulse imparted to the sphere by the surface, perpendicular to the surface will also have magnitude J.

If $μ = \frac{11}{20},$ sphere rolls with backward slipping

If $μ = \frac{13}{20},$ sphere rolls with forward slipping

answer is A, B, C.

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Detailed Solution

$J R \cos 37 - J' R = \frac{2}{5} m R^{2} ω \Rightarrow \frac{4 J}{5} - J' = \frac{2}{5} m V$ and $J' = m V$

$\Rightarrow \frac{4 J}{5} = \frac{7}{5} m v$

$J = \frac{7}{4} m v$

$\to b u t J' \leq μ J \Rightarrow m v \leq μ \frac{7}{4} m v$

$μ \geq \frac{4}{7}$

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