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Question

To a billiard ball of mass M and radius r a horizontal impulse is given passing through its centre of mass and it provides velocity u to its centre of mass. If coefficient of friction between the ball and the horizontal surface is μ, then after how much time ball starts pure rolling?

Joshi sir comment

for translational motion

0-μMg = Ma so a = -μg

so after time t velocity v = u--μgt

for rotational motion

about centre, μMgr = Iα = 2Mr²α/5

so α = 5μg/2r

so ω = 0 + αt

so ω = 5μgt/2r

for pure rolling

v = rω implies that u-μgt = r5μgt/2r

implies that u = 7μgt/2

so t = 2u/7μg

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