Inertia is a fundamental concept in mechanics. Moment of inertia is the inertia associated with a rigid body. It is used in several real-life situations. Let us look into the Parallel Axis Theorem Formula.

## Parallel Axis Theorem

Parallel axis theorem states that The moment of inertia of a body about an axis parallel to the body passing through its centre is the sum of moment of inertia of a body about the axis passing through the middle and product of the mass of the body times the square of the distance between the 2 axes

## Inertia

When we change the inertia of a body we encounter resistance. This resistance is known as inertia. The change can be of the sort to change the speed, or direction of motion of the body. Thus there is a tendency for bodies to keep moving in a straight line if no forces are acting upon them. This aspect forms the crux of Newton’s first law of motion, which is stated as follows in his book *Principia Mathematica:*

“The *vis insita*, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.”

## Parallel Axis Theorem Formula

Parallel axis theorem statement is as follows:

\(I = I_c + Mh^2\)

Where,

- I = moment of inertia of the body
- I
_{c}= moment of inertia about the centre - M = mass of the body
- h
^{2}= square of the distance between the two axes

## Parallel Axis Theorem Derivation

Let I_{c} be the moment of inertia of an axis which is passing through the centre of mass (AB from the figure) and I will be the moment of inertia about the axis A’B’ at a distance of h.

Consider a particle of mass m at a distance r from the centre of gravity of the body. Then, Distance from A’B’ = r + h

I = ∑m (r + h)^{2}

I = ∑m (r^{2} + h^{2} + 2rh)

I = ∑mr^{2} + ∑mh^{2} + ∑2rh

I = I_{c} + h2∑m + 2h∑mr

I = I_{c} + Mh^{2} + 0

I = I_{c} + Mh^{2}

The final formula is the formula for the moment of inertia.

## Solved Examples for Parallel Axis Theorem Formula

Q1: If the moment of inertia of a body along a perpendicular axis passing through its centre of gravity is 50 kg·m^{2} and the mass of the body is 30 Kg. What is the moment of inertia of that body along another axis which is 50 cm away from the current axis and parallel to it? Use Parallel Axis Theorem Formula

Solution: From parallel axis theorem,

I = IG + Mb2

I = 50 + ( 30 × 0.5^{2} )

I = 57.5 kg – m^{2}

Q2: Calculate the moment of inertia of a rod whose mass is 30 kg and length is 30 cm?

Solution: The parallel axis formula for a rod is given as,

I = (1/12) ML^{2}

plugging in the values we get

I = 0.225 Kg m^{2}.

Q3: Calculate the moment of inertia of a stick whose mass is 100 gm and length is 10 cm?

Solution: The parallel axis formula for a rod is given as,

I = (1/12) ML^{2}

plugging in the values we get

I = 0.0000833 Kg m^{2}.