Parallel axes theorem of moment of inertia helps in finding
the moment of inertia of a body about given axis when we know the moment of
inertia of the same body about its axis passing through center of gravity and
the perpendicular distance between the two parallel axes.

Moment of inertia of a body depend on its mass and distribution of mass about its axis of rotation.Moment of inertia changes with axis of rotation.With the change of axis of rotation, distance of each particle changes and hence moment of inertia also changes.

Measuring moment of inertia is a bit complex process. We need to measure the mass of each particle, distance of that particle from axis of rotation and square that distance. Further we need to multiply this mass of that particle with the square of the distance. This process has to be repeated for all the particles and we need to sum all of them. This can be done mathematically using a method called integration.

To avoid this complexity, we have a method called parallel axes theorem and perpendicular axes theorem. This theorems helps you in measuring the moment of inertia of a system when axis of rotation changes without using the integration process.

According Parallel axes theorem, moment of inertia of a body about a given axis is the sum of moment of inertia of the same body about an axis passing through center of gravity and the product of mass of the body with the square of the distance between two parallel axes between them.

According to perpendicular axes theorem, moment of inertia of a body about given axis is the sum of moment of inertia of the same body about two perpendicular axes passing through in the same point in the perpendicular plane.

These two theorems helps in identifying the moment of inertia of the body about different axis when we know the moment of inertia of the same body about a given axis.

In the following video lesson, it is demonstrated in detail about parallel and perpendicular axes theorem.

Moment of inertia can be mathematically defined as the sum of
product of mass of each particle with the square of the distance from axis of
rotation. It depends on the mass of the particle and distance of the particles
from axis of rotation. It changes with size of the body, shape of the body and
axis of rotation.

**Parallel Axes theorem**

Let there is a particle of mass

**m**at a particular distance of from axis of rotation. Let it is at a distance from the axis of rotation which is passing from center of gravity. Its moment of inertia can be measured as the product of mass of the body with the square of distance of separation. Let there is a parallel axis and there is a known perpendicular distance between the two axes. The moment of inertia of the particle is different from the moment of inertia of the same particle about an axis from center of gravity.
We can rewrite the moment of inertia of the particle about
the parallel axis in terms of the moment of inertia of the particle about the
moment of inertia of the axis of rotation about center of gravity.

To find the moment of inertia of the entire body, we have to
add the moment of inertia of the particles of all the particles. It can be
proved as per the definition.

**Perpendicular axes theorem**

Moment of inertia of a body about an axis passing through a
point is equal to the sum of moment of inertia of the same body about two
perpendicular axes passing through the same point.

This theorem helps in measuring the moment of inertia of a
body about a perpendicular axis.

Let us consider a particle of known mass in the body from the
point of consideration. It will have coordinates which represents its position
in the system. We can identify the moment of inertia of that particle as the
product of mass of the body with the square of the distance from axis of
rotation. By adding the same for all particles of the body, we can measure the
moment of inertia of the total body.

Now we can express this distance of the particle as the sum
of squires of the distance about the two perpendicular axes passing through the
same point. Hence it lead to the proof of the perpendicular axes theorem.

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