**Moment of Inertia **

According to Newton’s first law of motion, a body must continue to be in its own state of rest or uniform motion unless compelled by some external force. On the basis of this law, inertia is the inertness or inability of a body to change its state of rest or uniform motion by itself. It is a fundamental property of a matter.

Similarly, in rotational motion, a body rotating about an axis opposes any change desired to be produced in its state. This property of the body is called rotational inertia or moment of inertia.

_{1}, m

_{2}, m

_{3, }........ m

_{n}. Suppose the body be rotating about an axis YY’ at a distance of r

_{1}, r

_{2}, …..., r

_{n}as shown

_{1}r

_{1}

^{2}, m

_{2}r

_{2}

^{2}, m

_{3}r

_{3}

^{2 },……,m

_{n}r

_{n}

^{2 }. Then the moment of inertia I of the body is

_{1}r

_{1}

^{2 }+ m

_{2}r

_{2}

^{2 }+ m

_{3}r

_{3}

^{2 }+ ……. + m

_{n}r

_{n}

^{2}

_{i}r

_{i}

Hence, the moment of inertia is the sum of the product of the masses of the various particles and square of their perpendicular distances from the axis of rotation.

**Calculation of Moment of Inertia of rigid bodies**

_{m}= mx

^{2}….... (i)

^{ l/2}ʃ

_{-l/2}I

_{m} =

^{l/2}ʃ

_{-l/2}mx

^{2}

^{l/2}ʃ

_{-l/2}M/l x

^{2}dx

^{l/2}ʃ

_{-l/2}x

^{2}dx

^{2 }/ 12

Let us consider a thin and uniform rod of mass ‘m’ and length ‘l’ rotate about an axis YY' passing from its one end. Let us suppose WX be another axis that passes through the center of the rod.

_{m}= mx

^{2}…... (i)

^{ l/2}ʃ

_{-l/2}I

_{m} =

^{l/2}ʃ

_{-l/2}mx

^{2}

_{cm}=

^{l/2}ʃ

_{-l/2}M/l x

^{2}dx

_{cm}= M/l

^{l/2}ʃ

_{-l/2}x

^{2}dx

_{cm}= Ml

^{2 }/ 12

Therefore, this is the moment of inertia for the body when it is rotating about WX.

_{cm}+ mr

^{2}

^{2}/ 12 + Ml

^{2}/4

^{2}/ 3

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