Rotational Kinetic Energy

Amongst various branches of physics, classical mechanics forms one of the most interesting and intellectually demanding sections. Mechanics finds variety of applications in the construction and civil engineering projects. Broadly classified, linear mechanics and rotational mechanics are the two branches of mechanics. Kinetic energy, one of the most fundamental concepts of physics helps in understanding various finer dimensions of many concepts in higher order physics. In the linear motion or what we more aptly refer to as one dimensional motion, linear kinetic energy is given by the formula, K.E_linear = I/2mv², where, m = mass of the body and v = linear velocity.

The above formula is something about which most of us are aware. Analogous to the one dimensional motion in linear mechanics, there is rotatory motion in rotational mechanics. It is basically the study of motion of bodies and objects which follow rotatory motion or circular motion. Force in linear motion is equivalent to 'torque' in rotational mechanics. Similarly, mass in linear mechanics is analogous to moment of inertia in rotational mechanics. The beauty of physics lies in witnessing such a wonderful analogy of various interrelated concepts that prove the coherence and order in the formation of this Universe. So as we admire the fascinating field of physics, let us know what exactly is kinetic energy in rotational mechanics.

What is Rotational Kinetic Energy?
Every moving object stores in itself various forms of the energy and the most dominant form of energy a moving object stores is kinetic energy. In cases of rotational motion, this energy comes through rotation of the object along some specific axis of rotation. Flywheels, planets and stars rotate on their own axis which is known as spin rotation.

It is an integral part of the total kinetic energy of a system that is possessed by the virtue of motion of the object about an axis of rotation. Stated simply, the energy possessed by rotation of rigid body about any line through the center of mass, is nothing but rotational kinetic energy (E_r).

The Formula
The formula can easily be understood by considering the rotational motion aspect of the object rotating about an axis. The total energy of a body is the sum of translational kinetic energy (E_t) and rotational kinetic energy (E_r). The translational kinetic energy of an object is due to the translational motion of the object (in other words, motion of the center of mass) and rotational kinetic energy is due to rotation about the center of the mass. In essence,

K.E_total = E_t + E_r
That was the formula for total kinetic energy of a body having both rotational and translational motion. E_r, in fact, is dependent on three factors, angular velocity (ω) and moment of inertia (I). Using these two, the formula for rotational kinetic energy for an object moving around a fixed rotational axis is given by,

Rotational kinetic energy (E_r) = ½ x I x ω²

where I = moment of inertia
ω = angular velocity

The rotational kinetic energy of the Earth is around 2.14 x 10²⁹J. To calculate this, we use the formula and put in the corresponding values that are as follows:

I = moment of inertia of Earth = 8.04 x 10³⁷ kg - m²
ω = angular velocity of earth = 7.29 x 10^-5rad.s^-1

This was a summarized information on rotational kinetic energy that is used frequently in higher order mechanics and is used to solve numerical problems.