A billiard ball has mass M, radius R, and moment of inertia about the center of mass(sphere...so (2/5)MR^2).
The ball is sturck by a cue stick along a horizonal line through the ball's center of mass so that the ball initially slides with velocity vi. As the ball moves across the billiard table (coef of fric. mu_k), its motion gradually changes from pure translation through rolling with slipping to rolling without slipping.
a) Write an expression for linear velocity(v) of the center of the ball as a function of time while it is rolling with slipping.
b) Write an expression for the angular veolicty(omega) of the ball as a function of time while it is rolling with slipping.
c) Determine the time at which the ball begins to roll without slipping
d) When the ball is struck it acquires an angular momentum about the fixed point P on the surface of the table. During th esubsequent motion the angular momentum about point P remains constant despite the frictional force. Explain why this is so.
Net torque= I*alpha
torque=F cross R
when not slipping omega= v/r ; alpha = a/r
The Attempt at a Solution
here's my attempt...
F in this case is the force of friction
so Ff=Ma; Ff is mu_k(Mg)
so a = mu_k(g)
So I said answer is vi+mu_k(g)t
torque= I*alpha ;torque=F cross R
F is Ff which is mu_k(Mg), R is just R; so torque= mu_k(Mg)R
I divided the torque by moment of inertia(2/5)MR^2 to get the anuglar acceleration.
so alpha= (mu_k*g)/(0.4R)
omega_f=omega_i + alpha(t)
so my answer was (mu_k*g*t)/(0.4R)
its not slipping so alpha=a/R;
a/R = (mu_k*g)/R
alpha = (mu_k*g)/(0.4R)
I'm stuck here...don't know what to do...do I look for the time when alpha equals a/R?
d)I just said there are no external forces, and when no external forces are applied, momentums are always conseved.