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a thin spherical shell of radius R lying on a rough horizontally surface is hit sharply and horizontally by a cue.Where should it be hit so that the shell does not slip on the surface?
Comments (6)
firstly we assume that the force of the cue stick on the ball is large and over a very short period of time so that,we can ignore the frictional force acting on the ball during the time the cue stick is acting
Second, since rolling without slipping is to begin immediately, we assume the instantaneous acceleration of the center-of-mass, a, is equal to alpha*R.
Fsin(theta)*R = I*alpha
Fh = (2/3)MRa
a=F/M
h=(2/3)R
So we want to hit the cue ball at a point 2/3 R above the center. Hitting below this point causes the ball to slide some distance before rolling without slipping. Hitting above this point causes the cue ball to have top spin. It spins faster than it should and has to slow its rotation rate before rolling without slipping can begin.
let it be hit at a height 'h' above the centre (not along diameter)
under the action the impulsive force F from the angular momentum theorem the ball will acquire some angular velocity 'w' while leaving the cue
from the linear impulse momentum theorem for the ball during the course of impact
change in momentum along x-direction , delta Px =< Fx>dt
mv - 0 =Fdt-----------------equation 1>
now from angular impulse momentum theorem about an axis passing through the centre of mass of the ball and perpendicular to plane of figure (sorry not able to upload figure view the plane as you see in ur notebook when u draw the diagram of spherical ball)
dLz =TORQUE dt
Ic*w - 0= F *h* dt---------------equation 2>
using eqns 1> & 2>
it implies
w=m*v*h/Ic
use v=wr
1=m*r*h /(2/3mr^2)
2r/3=h
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