Question

A billiard ball with a mass of m and a radius of r is rolling down without friction along an inclined plane with a slope of Θ at h height from the floor.

(1) Find an inertial moment for the center of a billiard ball. Here the mass distribution of billiard balls is assumed to be uniform.

(2) What is the speed of a billiard ball when it reaches the floor?

((1)If you can't solve the question yourself, use inertia moment I=2/5*mr^2.)

Answer 1

Solution :

Part (a) Solution :

Inertial moment for the center of a billiard ball : I = (2/5) m r²

^.

Part (b) Solution :

Let the speed of the billiard ball when it reaches the floor is : v.

Then, Total kinetic energy of the billiard ball at the bottom of the inclined plane will be : KE_total = KE_rot + KE_tran

∴ KE_total = (1/2) I ω² + (1/2) m v²

∴ KE_total = (1/2) {(2/5) m r²} (v/r)² + (1/2) m v²

∴ KE_total = (1/5) mv² + (1/2) m v²

∴ KE_total = (7/10) m v²

^.

Now, According to the conservation of energy : KE_f = PE_i

∴ (7/10) m v² = m g h

∴ (7/10) v² = g h

∴ v² = (10/7) g h