Question

2. A bowling ball is thrown down an alley with initial velocity vo and no rotation around its center of mass. The ball is thrown in such a way that it initially slides before beginning to roll. The ball has mass m, coefficient of static friction Hls, and coefficient of sliding friction plk. Show that the velocity of the ball when it begins to roll without slipping is vo. (Hint: at first the ball is sliding with friction present, so energy isn't conserved, don't do that!)

Do not use energy conservation.
Because of friction, energy isn't conserved.

Answer 1

Even if friction is present the it pass through the contact point, sphere's weight and normal reaction also pass through the contact point hence net torque about that point is zero. Therefore angular momentum is conserved about that point.

Taking final velocity to be v and angular velocity to be ω

mv₀R = mvR + Iω

The bowling ball can be assumed to be a uniform sphere, hence I = 2/5(mR²)

mv₀ = mv + (2/5)mRω
v₀ = v + (2/5)(Rω)

For Pure Rolling v = Rω

v₀ = v + (2/5)v

v₀ = (7/5)v

v = (5/7)v₀