Critical Thinking

Moment the total amount of matter contained inside

Moment of inertia: While calculating the kinetic energy of a particle being in linear motion, we use the formula, ……………………………………(1) Where m and v are the mass and the velocity of the particle. In rotational motion though, the case is different. Kinetic energy calculation needs a new concept which we are going to discuss here. Let a body of mass m is rotating about an axis. The distance between the body and the axis pf rotation is r. Let  be the rotational velocity and V be the corresponding linear velocity. So, …………………………………………………(2) And its kinetic energy is given by eq(1) by putting the value of V using eq(2) as, ………………………..(3) If the particle is made up of n number of tiny particles, then, the kinetic energy of the ith particle having mass mi and distance from the axis of rotation ri is   Since the body is rotating as a whole,  will be same. considering all the particles, the kinetic energy will be,  The quantity in bracket is denoted as,  ,  Here, I is known as the moment of inertia of the body and this is that new concept. The kinetic energy of the body will now become ……………………………..(4) This the rotational kinetic energy of the body. If we compare eq(1) and eq(4), we can see that I is analogous to m. Hence, we can say that the role of mass in linear motion is played by moment of inertia in rotational motion and thus known as the rotational equivalent of mass. The dimension of MI is   As mass of a body resists any change is the state of motion in straight line, it is a measure of its inertia in linear motion. Since MI is the rotational equivalent of mass, MI also resists any change in the state of rotational motion and is the measure of inertia in rotational motion. MI also tells us about how the constituent particles are distributed in a body. We all know that mass is the total amount of matter contained inside the body an is a fixed quantity. In case of MI, it is not a fixed quantity but varies with distance from the axis of rotation and the orientation of the body about the axis of rotation. As MI is the measure of how the constituents are distributed inside the body, let us define radius of gyration.  The radius of gyration of a body about its rotational axis is defined as the distance from the axis of a point mass having mass equal to the mass of the whole body and as a consequence, its MI about the axis of rotation will be equal to that of the body.