JEE Plances Class Answered

Force balance: -  mg = R₁+µR₂

Normal force R₂ is unbalanced and makes the sphere to move right to left.

Torque due to the friction force µR₂ retarding the circular motion.

Hence µR₂×r = I α = (2/5) mr²×α ……………………………(1)

I is Mom. Of Inertia and α is angular retardation

Substitute for µ( µ = 1/5 )and simplify to get

R₂ = 2mrα …………………………………………………………(2)

Let the unbalanced horizontal force R₂ changes the linear velocity from v to V during the contact time t

Then R₂ = m(V-v)/t………………………………………..(3)

From (2)and (3)  m(V-v)/t = -2mrα

Negative sign on RHS is due to retardation

After simplification we get   (V-v) = -2rαt……………(4)

If initial angular velocity is ω and if angular velocity is zero after contact time t, then we can write

0 = ω – αt or ω = αt

Substitute for αt in equation (4) to get (V-v) = -2rω = -2v

Hence V = -v (negative sign indicates sphere moving from right to left)

If we take the magnitude of velocity immediately after collision, |V| = v