Kinetic Energy of a rolling body
The rolling body in an inclined plane has both rotational as well as translational motion as the body rotates as well as travels some distance in a straight line. The total kinetic energy of a rolling body can be found by adding its translational as well as rotational kinetic energy.
Total K.E = translational K.E + rotational K.E
Let’s consider a symmetric circular body (such as a cylinder or sphere) of mass and radius rolling down on an inclined plane as shown in the figure below. Let be the angle of inclination and a body starts to roll down from rest and reaches velocity after traveling displacement along the inclined plane with a vertical height of .
Translational Kinetic Energy (K.Etranslational) =
Rotational Kinetic Energy (K.Erotational) =
Hence, Total Kinetic Energy (K.E) =K.Etranslational +K.Erotational
But, (‘k’ being the radius of gyration of a body) and
This is the expression of Kinetic Energy of a rolling body in terms of the radius of gyration . But in terms of moment of inertia, the expression would be:
Acceleration of a rolling body
The kinetic energy gained by a rolling body after traveling displacement ‘s’ in an inclined plane is:
But the body loses height as it rolls down, hence losing its potential energy. The loss in potential energy is given by:
But from the figure,
Form the principle of conservation of energy,
gain in K.E = loss in P.E
If the body gains acceleration in that instant of time, then we can write:
Since the body starts to roll from a rest. We can write
This is a required expression for the linear acceleration gained by a rolling body.