Kinetic Energy and acceleration of a rolling body

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.E_{translational}) =

Rotational Kinetic Energy (K.E_rotational) =

Hence, Total Kinetic Energy (K.E) =K.E_{translational} +K.E_rotational

   

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.