19.11 Work and energy in rolling motion  (Page 2/3)

The static friction has dual role here. If “T” and “R” denotes translation and rotation respectively, then :

and

where “θ” is total angle covered during the motion. For rolling motion,

Putting the expression of angle in equation – 8,

$$\begin{array}{l}{W}_{\mathrm{fR}}=f_{S}Rx\frac{x}{R}=f_{S}x\end{array}$$

The work done by friction in translation and rotation are equal, but opposite in sign. The net work by friction is zero.

Thus, total work done by the external forces is :

We find that work by friction in rolling is zero. It is so because it does negative work in translation and equal positive work in rotation. There may be situations in which friction does positive work in translation and negative work in rotation. For example, if a body is initially given angular velocity greater than linear velocity required by equation of rolling, then the friction does the positive work in translation (accelerates translation) and negative work in rotation (decelerates rotation). Here also, net work by friction is zero in rolling.

Mechanical energy of rolling body

Mechanical energy comprises of potential and kinetic energy of the rolling body. These two forms of energy is described, in reference to rolling, as here :

Potential energy

We must have observed the conspicuous absence of reference to the potential energy in rotation. The reason is simple. The rotating body does not change its center of mass over a period of time in pure rotation. There is no change in the body – Earth configuration due to rotation. This is a valid approximation for a relatively small body such that its mass distribution with respect to Earth remains same.

If a body is rolling along a straight line on a horizontal surface, then there is no change in potential energy as COM of the rolling body remains at same vertical height from the ground. In the case of rolling along an incline, the COM of the rolling body changes elevation and as such there is change in the gravitational potential energy of the body – Earth system (Note that potential energy is always referred to a system – not to a single body like kinetic energy. When we assign potential energy to a body, the reference to Earth or other force field is implicit). It must be borne in mind that potential energy changes in rolling is on account of translation only.

where “h” is the change in vertical elevation.

Kinetic energy

Unlike potential energy, kinetic energy arises from both translation and rotation. The kinetic energy of rolling body is a positive sum of translational and rotational kinetic energies. The total kinetic energy of a rolling body is given by :

$$\begin{array}{l}K=K_T+K_R=\frac{1}{2}xM{V_C}^2+\frac{1}{2}xI{\omega }^2\end{array}$$

Looking at the expressions of translational and rotational kinetic energy, we find that we can convert the expression either in terms of linear or angular velocity, using equation of rolling,

$$\begin{array}{l}{v}_C=\omega R\end{array}$$

Total kinetic energy in terms of linear velocity,

Total kinetic energy in terms of angular velocity,