A fascinating problem that combines impulse, angular impulse and rolling: at what height must a billiard ball be struck with the cue so that it rolls without slipping from the very start, without skidding on the cloth?
The intuition is that the cue does two things at once. It gives the ball a translational impulse (which accelerates the centre of mass) and, since it strikes at a certain height above the table, it also gives an angular impulse about the point of contact (which sets it spinning). If you strike too low, the ball translates but doesn’t spin enough and skids forward; if you strike too high, it spins too much and skids backward. There is a magic height at which the two effects match perfectly and pure rolling begins immediately.
Imposing the translational impulse and the angular impulse about the point of contact, with the rolling condition , gives
You must strike at of the radius above the table, i.e. slightly above the centre of the ball (which sits at height ). It’s the secret every good billiard player knows by instinct: striking just above the centre sets the ball off in smooth rolling, without the initial skid that would make its trajectory unpredictable. The full working is in the dedicated worked exercise.
Connections
Topics: Dinamica rotazionale Concepts: Rotolamento · Momento angolare · Impulso Objects: Sfera che rotola
Related exercises: Esercizio svolto — altezza ottimale per la steccata · Esercizio svolto — sfera che rotola giù da un piano · Problema — Momento torcente rettangolare e L(t)