moment of inertia The moment of inertia of a rigid body is the measure of inertia of the body to an external torque.

Consider a rigid body rotating about an axis AB (see fig. m15). A particle at the point P of mass m , has the kinetic energy K_r ,

K_r = (1/2) m_i v_i²

where v_i is the linear velocity of the particle. To obtain the total kinetic energy we sum over all particles of the body.

K_r = S (1/2) m_i v_i²

Since v_i = w r_i

where w is the angular velocity of the particle,

K_r = (1/2)w ^2S m_i r_i²

= (1/2) I w ²

where I = S m_i r_i²

is defined as the moment of inertia of the body. K_r can also be written as

K_r = (1/2) M k² w ²

where M is the total mass of the body. The constant k is given by,

k = (1/M) S m_i r_i²

k is called the radius of gyration of the body. At the distance equal to the radius of gyration the whole mass of the body can be thought to be concentrated.

For rigid bodies with mass homogeneously distributed, the summation can be replaced by integration. For regular geometrical shapes the integration can be computed (see table MI).

Table MI. Moment of inertia of some rigid bodies.

There are two useful theorems that helps to find the moment of inertia of a rigid body about another axis if it is known for one axis.

Parallel axis theorem: If I is the moment of inertia of a body about any axis through its center of mass, and I is the moment of inertia about a parallel axis ,

I = I + m a²

where a is the perpendicular distance between the two axis, and m is the mass of the body.

Perpendicular axis theorem (only applicable to a lamina): If x and y are two axes mutually at right angles, in the plane of a lamina, and z axis is normal to the plane of the lamina, then

I _x + I_y = I_z

where I_x , I_y and I_z are respectively the moment of inertia about the axes x,y, and z.