# MOMENT OF INERTIA FORMULA

**DEFINITION OF MOMENT OF INERTIA** – **Moment of Inertia** of a rotating body is its property to oppose any change in its state of uniform rotation. If in a given rotational system particles of masses M_{1}, M_{2}, M_{3}, ………… be situated at normal distances R_{1}, R_{2}, R_{3}, ………… from the axis of rotation, then **moment of inertia** of the system about the axis of** rotation** is given by .

** FOR POINT MASS BODY MOMENT OF INERTIA IS**,

I=M_{1}R^{2}_{1}+M_{2}R^{2}_{2}+M_{3}R^{2}_{3}+ ……..=∑MR^{2}

For a rigid body having a continuous mass distribution , **moment of inertia** is

I=∫dm.r^{2}

**S.I unit of moment of inertia is m.kg ^{2}** .

It is neither a scalar nor a vector, i.e. it is __tensor.__

__Moment of inertia formula of parallel axes theorem is__

–**moment of inertia** of a body about a given axis I is equal to the sum of **moment of inertia** of the body about a parallel axis passing through its **centre of mass (COM)** I_{CM } and the product of mass (M) of body and square of normal distance d between the two axes.

__Moment of inertia formula for perpendicular axes theorem__

-The sum of **moment of inertia** of a plane laminar body about two mutually perpendicular axes lying in its plane is equal to its **moment of inertia** about an axis passing through the point of intersection of these two axes and perpendicular to the plane of laminar type body. If I_{X} and I_{Y} be **moment of inertia** of the body about two perpendicular axes in its own plane and I_{Z} be the **moment of inertia** about an axis passing through point O and perpendicular to the plane of plate, then

__MOMENT OF INERTIA FORMULA FOR REGULAR SHAPE OF OBJECTS –__

**i) Moment of inertia of uniform Ring of Mass M and Radius R**

About an axis passing through the centre and perpendicular to plane of ring

I=MR^{2 } , and about a diameter I=1/2MR^{2}

ii) **Moment of inertia of Uniform Circular Disc of Mass M and Radius R**

About an axis passing through the centre and perpendicular to the plane of disc

I=1/2MR^{2},

About a diameter,

I=1/4MR^{2}

iii) **Moment of inertia of Thin Uniform Rod of Mass M and Length L**

About an axis passing through its centre and perpendicular to the Rod ,

I=1/12ML^{2},

iv) **Moment of inertia of Uniform Solid Cylinder of Mass M, Length L and Radius R**

About its own axis , I=1/2MR^{2}

About an axis passing through its centre and perpendicular to its length,

I=M[L^{2}/12+R^{2}/4]

v) **Moment of inertia of Uniform Solid Sphere of Mass M and Radius R**

About its diameter , I=2/5MR^{2}

About its tangent, I=7/5MR^{2}

*** Radius of Gyration –
**

Radius of gyration of a given body about a given axis of rotation is the normal distance of a point from the axis, where if whole mass of the body is placed, then its moment of inertia will be exactly same as it has with its actual distribution mass.

Thus, radius of gyration,

K=√(I/M)

Or,

I=[(〖r_1〗^2+〖r_2〗^2+〖r_3〗^2+⋯〖r_4〗^2)/n]^(1/2)

S.I unit of Radius of gyration is ‘meter’

MOMENT OF INERTIA FORMULA IN PDF

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