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A solid sphere of mass m and radius R is placed on a plank of equal mass, which lies in a smoth horizontal surface.The sphere is given a sharp impulse as a result it stats slidding with velocity v.Find the time taken by the sphere to start pure roliing on the plank.the coefficient of friction between plank and sphere is k.
velocity of center of mass is v & angular velocity of sphere is 0 initially...
friction will act in backward direction ....now
f = kmg (f = friction force)
ma = kmg
a=-kmg (a is retardation of center of mass)
now , torque = fR = I(alfa) (alfa = angular accleration)
alfa = fR/I
= 5kg/2R (Isphere = 2MR2/5)
now after time t let velocity of ceneter of mass is V then
V = U + at (initial linear velocity is v)
V = v - kgt ............1
let at this time angular velocity os W then
W = Wo + (alfa)t (initial angular velocity is 0)
W = 5kgt/2R ............2
now if pure rolling has started then V = WR
so , v - kgt = 5kgt/2
t = 2v/7kg
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