#accelerated #pure #rolling|#rolling on a #rough #inclined #plane|#concept #friction in #Rolling

Till now we were discussing the uniform pure rolling in which v andwwere constants. Now, suppose
an external force is applied to the rigid body, the motion will no longer remain uniform. The condition
of pure rolling on a stationary ground is,
v = Rw
Thus, in addition to v = Rw at every instant of time there is one additional
condition, linear acceleration = R × angular acceleration or a = Ra for pure
rolling to take place. Here, friction plays an important role in maintaining
the pure rolling. The friction may sometimes act in forward direction,
sometimes in backward direction or under certain conditions it may be zero.
Here, we should not forget the basic nature of friction, which is a self
adjusting force (upto a certain maximum limit) and which has a tendency to
stop the relative motion between two bodies in contact and here the relative
motion stops when at every instant v = Rw. To satisfy this equation all the time, a = Ra equation
should also be satisfied. Let us take an example illustrating the above theory.
Suppose a force F is applied at the topmost point of a rigid body of radius
R, mass M and moment of inertia I about an axis passing through the
centre of mass. Now, the applied force F can produce by itself:
(i) a linear acceleration a and
(ii) an angular acceleration a.
If a = Ra, then there is no need of friction and force of frict
Differentiating this equation with respect to time, we have
dv
dt
R
d
dt
= .
w
or a = Ra