Theorem of Parallel axes gives the moment of inertia of a body about any axis in terms moment of inertia of a body about a axis parallel to first axis.
- We know that, the weight of the any body is due to its inertia. The moment of inertia of a body about any axis is nothing but the sum of product of mass of that each particle and square of the distance of each particle from that axis of rotation.
- That means, I = ∑ mi ri2
- The moment of inertia of the body depends on the mass, shape and size of the body.
- It also depends on the mass distribution of particles about the axis of rotation of the body.
- And also, on the position of the axis of rotation of the body.
- The SI unit of moment of inertia of the body is kg m2.
Statement:
Theorem of Parallel axes states that the moment of inertia of a body about any axis is equal to the sum of moment of inertia about a axis parallel to it and passing through centre of mass of the body and the product of the mass of the body & square of the perpendicular distance between those two parallel axes.
The following is neat labelled diagram which explains the parallel axes theorem.
- Thus, from figure we can write mathematical statement of parallel axes theorem as,
Io = Ic + Mh2
- Where, Io is the moment of inertia of a body about the axis passing through point O
- Ic is the moment of inertia of a body about the parallel axis passing through centre of mass C of the body.
- M is the total mass of the body
- And h is the perpendicular distance between the two parallel axis.
Applications:
- Parallel axis theorem can be used to find the moment of inertia of about an axis in case of any body.
- Parallel axes theorem can be used to find moment of inertia of a thin rod.
- It is used to find moment of inertia of a ring about a tangent in its plane.
- It is also used to find the moment of inertia of ring about a tangent perpendicular to its plane.
Example:
What is the moment of inertia of a ring about a tangent in its plane?
Solution:
Let Ic be the moment of inertia about the diameter of the ring as shown in figure.
And Io be the moment of inertia of a ring about the tangent which is parallel to diameter and the distance between two parallel axis is h = R.
Then by parallel axes theorem,
Io = Ic + Mh2
We know that, the moment of inertia of a ring about its diameter is Ic = 1/2MR2
Thus, Io = 1/2MR2 + MR2
Io = 3/2 MR2
Thus, the moment of inertia of a ring about tangent in its plane is 3/2MR2.