Perpendicular Axis Theorem And Parallel Axis Theorem

Inertia is one of the main concepts in physics. The parallel axis theorem and the perpendicular axis theorem help find the moment of inertia for bodies based on their type and plane. There are slight differences between these two theorems, which have separate use cases.

Let us go through all about the parallel axis theorem. We’ll cover the parallel axis theorem derivation, examples of this theorem, and the key differences from the perpendicular axis theorem. Let us start with a quick definition of the parallel axis theorem.

What is the Parallel Axis Theorem?

The parallel axis theorem states that the moment of inertia of the body under consideration about an axis parallel to the body passing through its centre is equal to the sum of the moment of inertia of the body about the axis passing through the middle and the product of the mass of the body and the square of the distance between the two axes.

Let m be the mass of the body,

h² be the square of the distance between the two axes,

Ic be the moment of inertia about the centre,

I be the moment of inertia of the body,

Hence, according to the parallel axis theorem, I = Ic + mh²

Derivation of the Parallel Axis Theorem:

It is easy to derive the parallel axis theorem.

Let Ic: the moment of inertia of the body with the axis passing through the centre of mass of the object, I: the moment of inertia about the axis at a distance h, m: mass of the particle, and r: distance of the particle from the centre of gravity of the body.

Hence, distance = r + h

I  = ∑m (r + h)²

I  = ∑m (r² + h² + 2rh)

I  = ∑mr² +  ∑mh² + ∑mrh

I  = Ic + h² ∑m + 2h∑mr

I  = Ic + Mh² + 0

I = Ic + Mh²

Solved Examples of the parallel axis theorem:

1. Find the moment of inertia of a rod having a mass of 40 kg and a length is 20 cm.

According to the parallel axis theorem, the moment of inertia of a rod is 1/12 ML²

Hence, putting the values, we get,

I = 1/12* 40 * 0.2* 0.2

I = 0.13 Kg m²

2. Find the moment of inertia of a stick having a mass of 500 gm and a length is 20 cm.

According to the parallel axis theorem, the moment of inertia of a rod is 1/12 ML²

Hence, putting the values, we get,

I = 1/12* 0.500 * 0.2* 0.2

I = 0.0017 Kg m²    

Difference between the perpendicular axis theorem and parallel axis theorem

Conclusion

Hence, it is easy to understand the main differences between the perpendicular axis theorem and the parallel axis theorem. Starting with the definition of the theorem, parallel axis theorem derivation, and proof, it becomes easy for the students to learn the basics of inertia.

The different examples of the parallel axis theorem make it easy to understand the different applications of this theorem. The quick difference table helps student clarify their concepts on parallel axis and perpendicular axis theorem. The quick questions further solve the possible queries of students without referring to other books.