Ques no 64
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
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