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Mass Moment of Inertia

It has been established since long that a rigid body is composed of small particles. If the mass of every particle of a body is multiplied by the square of its perpendicular distance from a fixed line, then the sum of these quantities (for the whole body) is known as mass moment of inertia of the body. It is denoted by I.

Consider a body of total mass m. Let it be composed of small particles of masses m1, m2, m3, m4, etc. If k1, k2, k3, k4, etc.,  are the distances from a fixed line, as shown in Fig. 1.4, then the mass moment of inertia of the whole body is given by

I = m1 (k1)2 + m2 (k2)2  + m3 (k3)2 + m4 (k4)2 + ….

If the total mass of a body may be assumed to concentrate at one point (known as centre of mass or centre of gravity), at a distance k from the given axis, such that

mk2 = m1 (k1)2 + m2 (k2)2 + m3 (k3)2 + m4 (k4)2 + …..
then——————————————————————- I = m k

The distance k is called the radius of gyration. It may be defined as the distance, from a given reference, where the whole mass of body is assumed to be concentrated to give the same value of I.

The unit of mass moment of inertia in S.I. units is kg-m2.

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