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Moment of Inertia, Radius of Gyration

Let us consider a rigid body rotating with a uniform angular velocity ω about an axis passing through O perpendicular to the plane of the paper as shown in the given figure.

Rotational kinetic energy and moment of inertia of a rigid body
The body is made up of a large number of particles. Consider a particle P of mass m1 at a distance r1 from the axis of rotation.

Let its linear velocity be υ1. The kinetic energy of this particle is


Similarly, the kinetic energies of all other particles (of the body) of a masses m2, m3, etc. are respectively , etc. where r2, r3, etc. are their respective distances from the axis of rotation.

The total kinetic energy (KE) of the body is given by


The quantity mr2 is the sum of the products of the masses of the various particles composing the body and the squares of their respective distances from the axis of rotation.

This quantity is called the moment of inertia of the body and its value depends upon the particular axis about which the body rotates and the way the mass is distributed in the body with respect to the axis of rotation.

Moment of inertia (MI) is usually designated by the symbol I.

In the two-dimensional rotation of the body about the z - axis, the kinetic energy is in terms of (x, y) components of r as


In the case of a body which does not consist of separate, discrete particles but has a continuous and homogeneous distribution of matter in it.


where dm is the mass of an infinitesimally small element of the body at a distance r from the axis of rotation.

The moment of inertia of a rigid body about a particular of axis may be defined as the sum of the products of the masses of all the particles constituting the body and the squares of their respective distances from the axis of rotation.

Physical Significance of Moment of Inertia

The above equation is the expression of the kinetic energy of the rotational motion of a body about an axis of rotation.

The kinetic energy of a body of mass m moving in a straight line with a velocity υ is


The linear velocity υ is an analogue of the angular velocity ω in rotational motion.

A comparison of the two equations suggests that the moment of inertia plays the same role in rotational motion as mass does in translational motion.

The inability of a body to change by itself its state of rest or uniform motion along a straight line is an inherent property of matter and is called inertia (Newton's first law of motion).

The greater the mass of a body, the more difficult it is for a force to move it or stop it if it is already moving.

The mass can be regarded as a measure of inertia for linear or translational motion.

In exactly the same way, a body free to rotate about an axis has a tendency to oppose any change in its state of rest or of uniform rotation.

In other words, it possesses inertia for rotational motion. i.e. it opposes the torque or the moment of the couple applied to change its state of rotation.

Since only a couple can oppose another couple, the nature of the inertia must be that of a moment.

The higher the moment of inertia of a body about an axis, the more difficult it is for a couple to rotate it or stop its rotation about the axis.

Thus moment of inertia can be regarded as a measure of inertia for rotational motion.

Unit of Moment of Inertia

The dimensions of moment of inertia I are (length)2 x (mass), i.e. ML2.

In the SI system the moment of inertia is, therefore, measured in kg m2.

It may be noted that the moment of inertia of a body about a given axis remains unchanged even when the direction of rotation about that axis is reversed,
Hence, it is a scalar quantity, once its axis of rotation is fixed.

Also the dimensions of K. E of rotation = Iω 2 are,
(dimensions of I) x (dimensions of ω2) = ML2 T-2 which are the same as those of work or energy.
Hence, KE is measured in joules.

Radius of Gyration

The radius of gyration of a body about its axis of rotation may be defined as the distance from the axis of rotation at which, if the entire mass of the body were concentrated, its moment of inertia about the given axis would be the same as with its actual distribution of mass.
It is usually denoted by the letter K.

If M is the mass of the body, its moment of inertia I in terms of its radius of gyration K can be written as
I = MK2

The expression for the radius of gyration of a body rotating about a given axis is derived as follows.

Consider a body of mass M consisting of n particles each of mass m, situated at distance r1, r2, r3, ...., rn from the axis of rotation.
The moment of inertia about the given axis is given by
I = m1 r12+ m2 r22 + m3 r32 + ...+ mn rn2
= = m(r12 + r22 + r32 +....+ rn2 ) ( m1 = m2 = ... m)


The quantity is clearly the mean (average) of the sum of the squares of distances of individual particles. Let us denote it by r2which is called the mean square distance. Thus
I = M ( M = m n)
MK2 = M
K2 =
or
i.e. K = root mean square distance.
Thus K is some kind of an average effective distance of the particles form the axis of rotation. So, the radius of gyration of a body about a particular axis of rotation is equal to the root mean square distance of its particles from the axis of rotation. K has the dimensions of L and is measured in metres.

Torque and Moment of Inertia

When a torque acts on a rigid body that is capable of rotation about an axis, it produces an angular acceleration in the body.

If the angular velocity of each particle of the body is ω, the angular acceleration is which is the same for all particles of the body but the linear acceleration varies with the distance r1, r2, r3, .... etc. of the particles from the axis of rotation. 

Consider a particle P of mass m1 rotating about an axis O with an angular velocity ω as shown in the figure.

If the distance of this particle from the axis of rotation is r1, its linear acceleration is
, where υ1 is its linear velocity.
υ1=r1ω
The force F1 acting on this particle is given by
F1 = Mass m1 x Acceleration of m1
The moment of force F1 about the axis of rotation is
Force (F1 ) x Distance ( r1 )
Similarly, the moments of the forces on other particles of masses m2, m3, …, mn, etc. are, etc. where r2, r3, ...., rn are their respective distances from O.

The sum of these individual moments is the total torque acting on the body, i.e,


where I = mr2 is the moment of inertia of the body about the given axis and is the angular acceleration produced by torque τ. Thus,
Torque = Moment of inertia ï Angular acceleration

Angular Momentum and Moment of Inertia

Angular momentum
In linear motion, an important property of a moving object is its linear momentum.

When an object rotates about an axis, its angular momentum plays an equally important part in its motion. The angular momentum of a body rotating about an axis is defined as the moment of linear momentum about that axis.

Consider a particle P of a rigid body rotating about an axis perpendicular to the (x, y) plane passing through the point O. Let m1 be the mass of particle P, r1 its distance form the axis of rotation and υ1 its linear velocity. Let ω be the angular velocity of the body about the given axis. Then the linear momentum of P has a magnitude p1 given by
P1 = Mass x Linear velocity
P1 = m1 υ1
P1 = m1r1ω  ( v1 = r1ω)
The magnitude of the angular momentum of P about C
L1 = Moment of momentum p1 about O
L1 = Momentum x perpendicular distance (moment arm)
L1 = p1 x r1
L1 = m1r12 ω
Similarly, the magnitude of angular momentum of other particles is m2r22ω, m3r32ω, .... mnrn2ω.
Therefore, the magnitude of total angular momentum of the body about the given axis is given by

where I =is the moment of inertia of the body about the given axis of rotation.

Illustrations of Conservation of Angular Momentum

A diver jumps from a high diving board keeping his legs and arms outstretched. His body has a certain momentum of inertia and a certain angular velocity about the centre of gravity of his body. To make a somersault he curls his body as shown in the figure. This decreases the moment of inertia and hence increases his angular velocity. He can then make more somersaults before entering the water below.

2. Dancers on skates can spin faster by folding their arms.

3. Figure shows a person standing on a turntable holding a pair of heavy dumb bells, one in each hand with his arms outstretched. The table is rotating with a certain angular frequency. The person suddenly pushes the weights towards his chest as shown in Fig.  The speed of rotation is found to increase considerably.

A driver turning a somersault

Conservation of angular momentum If r1 > r2, then ω 1 > ω 2

When a ballet dancer leaps across the stage in a grand jet, she raises her arms and stretches her legs out horizontally as soon as her feet leave the stage. These actions shift her center of mass upward through her body.

Although the shifting center of mass faithfully follows a parabolic path across the stage, its movement relative to the body decreases the height that would be attained by the head and torso in a normal jump. The result is that the head and torso follow a nearly horizontal path.




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