MOMENT OF INERTIA | Radius of gyration or Swing radius | Kinetic energy of a rotating body | Angular momentum | Torque | perpendicular axis theorem | parallel axis theorem

Define moment of inertia:

Moment of inertia

Definition (1) – For a single particle the moment of inertia about an axis is the product of mass of the particle and the square of the distance from the axis i.e.

For an extended body,

Definition (2):

Moment of inertia of a body about an axis is numerically equal to the angular momentum with unit angular velocity i.e.

———————- when w =1 then J=I

Definition (3):

when w=1 then 2E = I   i.e.

Moment of inertia of any body about any axis numerically equal to the twice the kinetic energy of the body with unit angular velocity rotating about that axis.

Definition (4):

when dw/dt = 1 then

Moment of inertia of a body about any axis numerically equal to the torque rotating with unit angular acceleration about that axis.

We know

Where M is the mass of the whole body

k is the radius of gyration of the body about the axis whose dimension is dimension of length.

Definition –

If we consider the whole mass of the body is concentrated to a point such that  Moment of Inertia of this point mass about the axis is equal to the Moment of Inertia  of the whole body about the axis, then the point from the axis of rotation is called the radius of gyration.

Kinetic energy of a rotating body:

Let us consider a rigid body is rotating about a fixed axis AB with uniform angular velocity w. We imagine the body is made up with a large number of particles of masses m1, m2, m3,——– situated at a distance r1,r2,r3,——- respectively from the axis of rotation.

Linear velocity of particle of m1,

Kinetic energy of particle of mass m1

Thus kinetic energy of the whole body

where ‘I’ is the moment of inertia of the body about that axis.

Angular momentum for a rotating body-

Angular momentum

The moment of momentum of body about an axis is known as

Let us consider a rigid body is rotating about a fixed axis As with uniform angular velocity w. We imagine that the body is made up with a large number of particles of masses m1, m2, m3,——— situated at a distance r1,r2,r3——– respectively from the axis of rotation.

Linear velocity of particle of mass m1,

Linear momentum of particle of mass m1

Moment of linear momentum of the particle of m1

= r1 × linear momentum

Moment of momentum of the whole body

where I = moment of the body

Torque of a rotating body :

Torque : Torque is moment of linear force.

Let as consider a rigid body is rotating about a fixed axis AB with uniform angular velocity w. We imagine the body is made up with a large number of particles of masses m1,m2,m3—— situated at distances r1,r2,r3—-respectively from the axis of rotation.

Linear acceleration of the particle of mass m1,

linear force acting on the particle of mass m1

Moment of linear force acting on the particle of mass m1

Moment of linear force acting on the whole body i.e. torque

Prove that rate of change of angular momentum of a body is equal to torque–

If I be the angular momentum of inertia of a body rotating with angular velocity w about an axis.

Then torque acting on the body,

————(1)

and angular momentum

J = Iw————-(2)

State and proof perpendicular axis theorem ;

Statement -The moment of inertia of a laminar body (2D) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia about two axes mutually at right angles to each other and lying on the plane of the Lamina.

If the lamina lies in X-Y plane and Moment of inertia of the body about X-axis = Ix

Moment of inertia of the body about Y-axis = Iy

Moment of inertia of the body about Z-axis = Iz

then

Proof:

Let a lamina lies in X-Y plane. Let P(x,y)  be the any point on the lamina having mass m.Then

Thus

(moment of inertia about z- axis)

Thus

Note:- For three dimensional body:

The sum of the Moment of inertia of a three dimensional body about three mutually perpendicular axes which lies in the body is equal to twice the Moment of inertia of the body about the origin of that coordinate system i.e.

State and proof parallel axis theorem :

Statement : The moment of inertia of a body (2D) about any axis is equal to the moment o inertia of the body about a parallel axis through its centre of mass plus mass of the body  and the square of the distance between two axes i.e.

Proof : Let C be the centre of mass of the lamina. We want to calculate centre of mass of the body of mass M about an axis AB at a distance h from centre of mass. Let us consider an elementary  particle of mass m at P (x,y) at a distance r from centre of mass axis .

Moment of Inertia of the particle about AB axis =

Moment of Inertia of the whole body about AB axis

i.e.

so, it is proved that

Where

mass of the whole body

Icm =  Moment of Inertia of the body about centre of mass axis.

sum of the moments of all particles of a body about centre of mass is zero.