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
r^{2}

^{.}

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 ω^{2} + (1/2) m
v^{2}

∴ KE_{total} = (1/2) {(2/5) m r^{2}}
(v/r)^{2} + (1/2) m v^{2}

∴ KE_{total} = (1/5) mv^{2} + (1/2) m
v^{2}

∴ KE_{total} = (7/10) m v^{2}

^{.}

Now, According to the conservation of energy : KE_{f} =
PE_{i}

∴ (7/10) m v^{2} = m g h

∴ (7/10) v^{2} = g h

∴ v^{2} = (10/7) g h

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