Moment Of Inertia

We have already mentioned that we are developing the study of rotational motion parallel to the study of translational motion with which we are familiar. We have yet to answer one major question in this connection. 

What is the analogue of mass in rotational motion? 

We shall attempt to answer this question in the present section. To keep the discussion simple, we shall consider rotation about a fixed axis only. Let us try to get an expression for the kinetic energy of a rotating body. We know that for a body rotating about a fixed axis, each particle of the body moves in a circle with linear velocity given by $v_i=\omega r_i$. (Refer to Fig. 1). For a particle at a distance from the axis, the linear velocity is $v_i=\omega r_i$. The kinetic energy of motion of this particle is

$k_i=\frac{1}{2}m_i v_i^2=\frac{1}{2}m_ir_i^2 \omega^2$

where $m_i$ is the mass of the particle. The total kinetic energy K of the body is then given by the sum of the kinetic energies of individual particles,

$K=\Sigma_{i=1}^{n}k_i=\frac{1}{2}\Sigma_{i=1}^{n}(m_ir_i^2\omega^2)$

Here n is the number of particles in the body. Note ωis the same for all particles. Hence, taking ω out of the sum,

$K=\frac{1}{2}\omega^2 (\Sigma_{i=1}^{n}m_ir_i^2)$

We define a new parameter characterising the rigid body, called the moment of inertia I , given by

$I=\Sigma_{i=1}^{n}m_ir_i^2$

With this definition,

$K=\frac{1}{2}I\omega^2$

Note that the parameter I is independent of the magnitude of the angular velocity. It is a characteristic of the rigid body and the axis about which it rotates.

Compare Eq. (7.35) for the kinetic energy of a rotating body with the expression for the kinetic energy of a body in linear (translational) motion,

$K=\frac{1}{2}mv^2$

Here, m is the mass of the body and v is its velocity. We have already noted the analogy between angular velocity ω (in respect of rotational motion about a fixed axis) and linear velocity v (in respect of linear motion). It is then evident that the parameter, moment of inertia I, is the desired rotational analogue of mass in linear motion. In rotation (about a fixed axis), the moment of inertia plays a similar role as mass does in linear motion.

We now apply the definition Eq. (7.34), to calculate the moment of inertia in two simple cases.

(a) Consider a thin ring of radius R and mass M, rotating in its own plane around its centre with angular velocity ω. Each mass element of the ring is at a distance R from the axis, and moves with a speed Rω. The kinetic energy is therefore,

$K=\frac{1}{2}mv^2=\frac{1}{2}MR^2\omega^2$

Comparing with Eq. (7.35) we get $I=MR^2$ for the ring.

Fig.1

(b) Next, take a rigid rod of negligible mass of length of length l with a pair of small masses, rotating about an axis through the centre of mass perpendicular to the rod (Fig. 1). Each mass M/2 is at a distance l/2 from the axis. The moment of inertia of the masses is therefore given by

$(\frac{M}{2})(\frac{l}{2})^2+(\frac{M}{2})(\frac{l}{2})^2$

Thus, for the pair of masses, rotating about the axis through the centre of mass perpendicular to the rod

$I=\frac{Ml^2}{4}$

Table 1 simply gives the moment of inertia of various familiar regular shaped bodies about specific axes. (The derivations of these expressions are beyond the scope of this textbook and you will study them in higher classes.)

Table 1 Moments of inertia of some regular shaped bodies about specific axes

As the mass of a body resists a change in its state of linear motion, it is a measure of its inertia in linear motion. Similarly, as the moment of inertia about a given axis of rotation resists a change in its rotational motion, it can be regarded as a measure of rotational inertia of the body; it is a measure of the way in which different parts of the body are distributed at different distances from the axis. Unlike the mass of a body, the moment of inertia is not a fixed quantity but depends on distribution of mass about the axis of rotation, and the orientation and position of the axis of rotation 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 1 that in all cases, we can write $I=Mk^2$, where k has the dimension of length. For a rod, about the perpendicular axis at its midpoint, $k^2=\frac{L^2}{12}$, i.e $k=\frac{L}{\sqrt{12}}$ 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 gyration. The radius of gyration of a body about an axis may be defined as the distance from the axis of a mass point whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis.

Thus, the moment of inertia of a rigid body depends on the mass of the body, its shape and size; distribution of mass about the axis of rotation, and the position and orientation of the axis of rotation.

From the definition, Eq. (7.34), we can infer that the dimensions of moments of inertia are $ML^2$ and its SI units are kg.m$^2$.

The property of this extremely important quantity I, as a measure of rotational inertia of the body, has been put to a great practical use. The machines, such as steam engine and the automobile engine, etc., that produce rotational motion have a disc with a large moment of inertia, called a flywheel. Because of its large moment of inertia, the flywheel resists the sudden increase or decrease of the speed of the vehicle. It allows a gradual change in the speed and prevents jerky motions, thereby ensuring a smooth ride for the passengers on the vehicle.