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 m and radius r rolling down on an inclined plane as shown in the figure below. Let \theta be the angle of inclination and a body starts to roll down from rest and reaches velocity v after traveling s displacement along the inclined plane with a vertical height of h.

Translational Kinetic Energy (K.Etranslational) = \frac{1}{2}mv^2

Rotational Kinetic Energy (K.Erotational) = \frac{1}{2}I\omega^2

Hence, Total Kinetic Energy (K.E) =K.Etranslational +K.Erotational

    \[\text{K.E} = \frac{1}{2}mv^2 + \frac{1}{2}I \omega^2 \]

But, I = mk^2 (‘k’ being the radius of gyration of a body) and \omega = \frac{v}{r}

    \[\text{K.E} = \frac{1}{2}mv^2 + \frac{1}{2}mk^2 \left( \frac{v}{r} \right)^2 \]

    \[\text{K.E} = \frac{1}{2}mv^2 + \frac{1}{2}mk^2 \frac{v^2}{r^2} \]

    \[ \boxed{ \text{K.E} = \frac{1}{2}mv^2 \left( 1 + \frac{k^2}{r^2}\right)} \]

This is the expression of Kinetic Energy of a rolling body in terms of the radius of gyration k. But in terms of I moment of inertia, the expression would be:

    \[ \boxed{ \text{K.E} = \frac{1}{2} v^2 \left( m + \frac{I}{r^2} \right) } \]

Acceleration of a rolling body

kinetic energy of rolling body
fig: Kinetic energy of rolling body

The kinetic energy gained by a rolling body after traveling displacement ‘s’ in an inclined plane is:

    \[ \text{K.E} = \frac{1}{2}mv^2 \left( 1 + \frac{k^2}{r^2}\right) \]

But the body loses height as it rolls down, hence losing its potential energy. The loss in potential energy is given by:

    \[\text{P.E} = mgh\]

But from the figure, h = s \sin\theta

    \[ \text{P.E} = mgs \sin\theta\]

Form the principle of conservation of energy,

gain in K.E = loss in P.E

    \[ \frac{1}{2}mv^2 \left( 1 + \frac{k^2}{r^2}\right)  = mgs \sin\theta \]

    \[ v^2 \left( 1 + \frac{k^2}{r^2}\right)  = 2gs \sin\theta \]

    \[ v^2   = \frac{2gs \sin\theta}{\left( 1 + \frac{k^2}{r^2}\right)} \]

If the body gains acceleration a in that instant of time, then we can write:

    \[v^2 = u^2 + 2as\]

Since the body starts to roll from a rest. We can write u = 0

    \[v^2 = 2as\]

    \[\frac{2gs \sin\theta}{\left( 1 + \frac{k^2}{r^2}\right)} = 2as\]

    \[\boxed{ a = \frac{g \sin\theta}{\left( 1 + \frac{k^2}{r^2}\right)} } \]

This is a required expression for the linear acceleration gained by a rolling body.

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