of translation
of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an
inclined plane of a height h is given by
using dynamical consideration (i.e., by
consideration of forces and torques). Note k is the radius of gyration of the
body about its symmetry axis, and R is the radius of the body. The body starts
from rest at the top of the plane.
Answer:
Velocity attained by a body rolling down an inclined plane.
Consider a body of mass M and radius R rolling down a plane inclined at an
angle 
with the horizontal,
as shown in Fig. 7.107. It is only due to friction at the line of contact that
body can roll without slipping.
The centre of mass of the body moves in
a straight line parallel to the inclined plane.
The external forces on the body
are
(i) The weight Mg acting
vertically downwards.
(ii) The normal reaction N of
the inclined plane.
(iii) The force of friction
acting up the inclined plane.
Fig. 7.107
Fig, 7.107 A body rolling
without slipping on an inclined plane.
Let a be the downward
acceleration of the body. The equations of motion for the body can be written
as
As the force of friction 
provides the
necessary torque for rolling, so
or 
where k is the radius of
gyration of the body about its axis of rotation. Clearly
or 
Let h be height of the inclined plane
and s the distance travelled by the body down the plane. The velocity 
attained by the
body at the bottom of the inclined plane can be obtained as follows:
or 
or 
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