A billiard ball, a hard sphere of uniform density, has mass m and radius r. It r

Question

A billiard ball, a hard sphere of uniform density, has mass m and radius r. It rolls without slipping on a felt-lined billiard table. The ball moves directly towards the edge of the table, and hits the overhang (see Fig 1.1 and Fig 1.2) Determine the height h of the overhang, such that the ball rolls without slipping immediately after the collision. State and explain any additional assumptions required. You may wish to note that the moment of inertia for a thin spherical shell of mass M and radius R about an axis through its centre is MR2) 5 marks] Figure 1.1: Top view of billiard ball rolling towards the edge of the table Figure 1.2: Side view of billiard ball rolling towards the edge of the table

Explanation / Answer

Torque about the center of mass of the ball after hitting the overhang will be ,

Tau = F.(h-R) ------------(1)

where F is the force imparted by the overhang to the ball upon collision. This force will generate acceleration resulitng ball to return back.

F = maCM ----(1)

And we also know torque to be Tau = Icm*alpha ----(2)

No slipping condition implies

aCM = R*alpha

Substituting from (2) and (3), we get

F/m = R*Tau / ICM

We know Tau from (1) and given ICM = 2/3 * mR²

F/m =R* F*(h-R) / (2/3 * mR²)

Solving we get,

h = 5R/3

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