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# Moment of Inertia of a Rotating Body

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Explain the concept of Moment of Inertia of a rotating body with examples.

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Understanding Rotational Motion
(Moment of Inertia)

Moment of Inertia to rotational motion is as mass is to translational motion. Hence, before we consider the concept of Moment of Inertia with reference to rotational motion, let's review and redefine the concept of mass with reference to translational motion.

Mass is generally defined in two ways viz. it is the measure of the amount of matter possessed by an object and alternatively it is the property of matter due to which an object offers resistance to any change in the state of its motion. Apart from these definitions we can define mass in alternative ways. For example, we know that the kinetic energy E possessed by an object moving with a velocity v is given by ½ mv^2 where m is the mass of the object. Hence, we can define mass of an object as that entity whose one half value multiplied by the square of the object's translational velocity gives the kinetic energy possessed by the object.

Further, from the fact that the quantity of motion possessed by a moving object (also called its momentum) is given by the product of its mass and velocity, mass can also be defined as that entity which multiplied by the velocity of the object gives the momentum of the object.

In the case of an object undergoing rotational motion about an axis with a rotational (or angular) speed ω (defined as the angle swept by a straight line drawn from the axis of rotation to any point on the object expressed in radians/sec), we can define its Moment of Inertia (I) as an entity whose one half value multiplied by the square of the rotational speed gives the rotational kinetic energy possessed by the object.

Rotational kinetic energy = ½ Iω^2

Using this definition of Moment of Inertia lets derive a general expression for the Moment of Inertia of any object. Fig. shows a generalised object rotating about the axis of rotation PQ with an angular speed ω (rad/sec).

Every material object can be visualised as comprising of an extremely large number of extremely small mass particles arranged to give the object its distinctive shape. Let there be n particles of masses m_1, m_2, ...... ,m_n. Then, total mass of the object M = m_1+m_2+ .... +m_n ...(1)
Let us consider one such particle located at a distance r from the axis of rotation PQ as shown.

Translational speed v of the mass m is related to its rotational speed ω and r as: v = ωr ...(2)
Hence, kinetic energy (translational) of mass m = KE_m = ½ mv^2 ...(3)

Substituting for v from (2) in (3) we get: KE_m = ½ m(ωr)^2 = ½ mr^2ω^2 ...(4)

As noted earlier, the object comprises n number of particles. Let these n particles be located at distances r_1, r_2, r_3, ......, r_n from the axis of rotation. Then, the kinetic energies of each of these particles are given by (using equation 4): ½ m_1r_1^2ω^2, ½ m_2r_2^2ω^2, ......., ½ m_nr_n^2ω^2.

Total KE of the object = KE_M = ½ m_1r_1^2ω^2 + ½ m_2r_2^2ω^2 + ....... + ½ m_nr_n^2ω^2

=> KEM = ½ (m_1r_1^2 + m_2r_2^2 + ....... +m_nr_n^2 )ω^2

Multiplying numerator and denominator by M, total mass of the object we get:

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