Let us assume that the mass of the ball is M and its radius is R.
The ball on its release begins to slide in the direction in which it has been
thrown. Therefore, it will experience a frictional force that will oppose its
motion.
As the ball is sliding, the frictional force will be kinetic and let it be
and its
direction will be as shown in the figure. The other forces that act on the ball
are the weight
acting at
its centre of mass in the vertical direction and the normal force acting in the
vertical direction at the point of contact of the ball with the surface of the
alley way. The frictional force
will
decelerate the sliding motion and it will also exert a torque about the centre
of mass of the ball that will produce angular acceleration in the ball about an
axis passing through the centre of the ball and perpendicular to the plane of
the diagram.
The weight
and the
normal force will not contribute to torque as their moment arm is zero. When the
ball is viewed in its centre of mass frame it will appear to rotate with angular
velocity that will be changing with time as long as the ball experiences the
torque,
. A stage
will come in the motion of the ball when the velocity of the point of contact of
the ball with the surface will become zero. At this stage the force of friction
will cease and the ball will no longer slide and rotate at the same time and
will start rolling without slipping.
We will now set up the equation of motion that determines the sliding speed
of the ball and its angular speed as function of time. Let
be the magnitude of the sliding deceleration. It will be given by the Newton’s
second law of motion
.
The speed with which the ball will be sliding at time
after it has been released with speed
is given
by the equation of motion of a decelerating body
.
The moment of inertia of a spherical ball of radius R and mass M
is
. The
angular acceleration
of the
ball is related to the torque
and the
moment of inertia by the counterpart of the Newton’s second law of motion for
rotational motion

The angular speed
as a
function of time will be
. Or
. The
speed of the point of contact of the rotating ball with the surface as measured
from its centre of mass will be
. The
ball will stop sliding and begin to roll without slipping at time
,
when

The reduced sliding speed at
when the
ball will stop sliding and begin rolling without slipping will be