# MOMENT OF INERTIA FORMULA

DEFINITION OF MOMENT OF INERTIAMoment 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 M1, M2, M3, ………… be situated at normal distances R1, R2, R3, ………… 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=M1R21+M2R22+M3R23+ ……..=∑MR2

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

I=∫dm.r2

S.I unit of moment of inertia is m.kg2 .

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) ICM 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 IX and IY be moment of inertia of the body about two perpendicular axes in its own plane and IZ 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=MR2 , and about a diameter   I=1/2MR2

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=1/2MR2,

I=1/4MR2

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=1/12ML2,

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

About its own axis ,               I=1/2MR2

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

I=M[L2/12+R2/4]

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

About its diameter  ,       I=2/5MR2

About its tangent,           I=7/5MR2

* 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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