MathDB

A uniform pool ball of radius

r

and mass

m

begins at rest on a pool table. The ball is given a horizontal impulse

J

of fixed magnitude at a distance

\beta r

above its center, where

-1 \le \beta \le 1

. The coefficient of kinetic friction between the ball and the pool table is

\mu

. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass

m

and radius

r

I_{cm} = \frac{2}{5}mr^2

.)
[asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.8)); draw((-3,-1)--(3,-1)); draw((-2.4,0.1)--(-2.4,0.6),EndArrow); draw((-2.5,0)--(2.5,0),dashed); draw((-2.75,0.7)--(-0.8,0.7),EndArrow); label("

J

",(-2.8,0.7),W); label("

\beta r

",(-2.3,0.35),E); draw((0,-1.5)--(0,1.5),dashed); draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow); label("

r

",(1.75,-0.5),E); [/asy]
(a) Find an expression for the final speed of the ball as a function of

J

m

, and

\beta

.
(b) For what value of

\beta

does the ball immediately begin to roll without slipping, regardless of the value of

\mu

Problem Statement