1 A horizo The ball leaves the cue R is the ntal impulse exe

(1) A horizo The ball leaves the cue R is the ntal impulse exerts strongly on a billiard ball at a distance h above its center radius of the ball. (a) After a while, the ball reaches a final pure rolling motiorn with initial speed Vo and angular speed oo that Vo=Rao /2, where due to friction, find the speed at that moment and (b) how high above the center does the cue strike? (25 points)

Solution

Given the intially the ball rotates faster than moves front hence there will be a retarding torque due to friction.

Hence the speed of the center of mass will increase and angulr velocioty will decrease

Conserving the angular momentum from the surface as the friction force acts from the surface.

I_cm w + mv_cm r_cm = contant

2/5mR² w_o + mvRw_o/2 R = 2/5mR² w + mwR²

9/10w_o = 7/5w

w = 9/14 w_o

v = 9/14R w_o

_{b) Let\'s say an impulse of P was given to the ball}

P *h = 2/5 mR² w_o

P = mw_oR/2

Hence

mw_oR/2 *h = 2/5 mR² w_o

h = 4/5 R