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SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 115 (a) Consider a thin ring of radius R and mass change in its rotational motion, it can be M, rotating in its own plane around its centre regarded as a measure of rotational inertia of with angular velocityω . Each mass element the body; it is a measure of the way in which of the ring is at a distance R from the axis, different parts of the body are distributed at and moves with a speed R ω . The kinetic different distances from the axis. Unlike the energy is therefore, mass of a body, the moment of inertia is not a 1 1 fixed quantity but depends on distribution of K = M υ2 = MR 2ω2 mass about the axis of rotation, and the 2 2 orientation and position of the axis of rotation Comparing with Eq. (6.35) we get I = MR 2 for the ring. with respect to the body as a whole. As a measure of the way in which the mass of a rotating rigid body is distributed with respect to the axis of rotation, we can define a new parameter, the radius of gyration. It is related to the moment of inertia and the total mass of the body. Notice from the Table 6.1 that in all 2 cases, we can write I = Mk , where k has the dimension of length. For a rod, about the perpendicular axis at its midpoint, 2 2 k = L 12, i.e. k = L 12 . Similarly, k = R/2 for the circular disc about its diameter. The length k is a geometric property of the body and axis of rotation. It is called the radius of Fig. 6.28 A light rod of length l with a pair of masses rotating about an axis through gyration. The radius of gyration of a body the centre of mass of the system and about an axis may be defined as the distance perpendicular to the rod. The total mass from the axis of a mass point whose mass is equal to the mass of the whole body and whose of the system is M. moment of inertia is equal to the moment of (b) Next, take a rigid rod of negligible mass of inertia of the body about the axis. length of length l with a pair of small masses, Thus, the moment of inertia of a rigid body rotating about an axis through the centre of depends on the mass of the body, its shape and mass perpendicular to the rod (Fig. 6.28). size; distribution of mass about the axis of Each mass M/2 is at a distance l/2 from rotation, and the position and orientation of the the axis. The moment of inertia of the masses axis of rotation. is therefore given by From the definition, Eq. (6.34), we can infer (M/2) (l/2) + (M/2)(l/2)2 that the dimensions of moments of inertia are Thus, for the pair of masses, rotating about 2 2 the axis through the centre of mass ML and its SI units are kg m . The property of this extremely important perpendicular to the rod quantity I, as a measure of rotational inertia of I = Ml / 4 Table 6.1 simply gives the moment of inertia of the body, has been put to a great practical use. various familiar regular shaped bodies about The machines, such as steam engine and the automobile engine, etc., that produce rotational specific axes. (The derivations of these expressions are beyond the scope of this motion have a disc with a large moment of textbook and you will study them in higher inertia, called a flywheel. Because of its large classes.) moment of inertia, the flywheel resists the As the mass of a body resists a change in its sudden increase or decrease of the speed of the state of linear motion, it is a measure of its inertiahicle. It allows a gradual change in the speed in linear motion. Similarly, as the moment of and prevents jerky motions, thereby ensuring inertia about a given axis of rotation resists a a smooth ride for the passengers on the vehicle. Reprint 2026-27