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19. Problem 12.35 (HR) A bowling ball is thrown down the alley in such a way that it
slides with a speed
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initially
without rolling. We have to prove that it will roll without any sliding when its
speed falls to
. The
transition from pure sliding to pure rolling is gradual, so that both sliding
and rolling takes place during this interval.
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Solution: Click For PDF Version 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
The weight
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
The speed with which the ball will be sliding at time
The moment of inertia of a spherical ball of radius R and mass M
is
The angular speed
The reduced sliding speed at
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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
. 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.
be the magnitude of the sliding deceleration. It will be given by the Newton’s
second law of motion
.
after it has been released with speed
.
. The
angular acceleration
of the
ball is related to the torque
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
when the
ball will stop sliding and begin rolling without slipping will be