Parallel Axis Theorem of a Rod

Physics encompasses understanding the workings of the universe. Physics covers all kinds of movements, from our physical actions to the earth’s movement. The concept of the moment of inertia, governed by its two main principles, explains the various movements occurring in the universe. The two principles are the parallel axis theorem and the perpendicular axis theorem. These two theories calculate the moment of inertia of objects in motion. 

What Is the Moment of Inertia?

The moment of inertia is the ability of resistance an object can apply while it is in motion. When an object is in motion, the resisting force is the moment of inertia. The moment of inertia is determined as the added value of the mass of an object upon which the moment of inertia acts. The moment of inertia is calculated in the SI unit of kg.m2

What Is the Parallel Axis Theorem?

The parallel axis theorem states, “The moment of inertia of a body about an axis parallel to the body passing through its centre is equal to the sum of moment of inertia of the body about the axis passing through the centre and the product of the mass of the body times the square of the distance between the two axes.” [1]

In the above statement, the moment of inertia refers to the object’s resistance in moving along the angular acceleration. The parallel axis theorem formula is expressed as:

I = Ic + MH2

In this formula,

l refers to the moment of inertia exerted by the body

lc is the moment of inertia present in the middle or the centre

M is the mass of the body in the moment of inertia 

H2 refers to the distance of the square when measured between the two axes. 

This formula of the theorem of parallel axis can be derived as stated below:

Let us consider lc as the moment of inertia present in the centre or the axis. Let this moment of inertia travel through the mass centre, which is taken as AB. Let l be the moment of inertia for A’B’. The distance here is measured and represented as h.

If we take an object with mass m, which is at a distance of r from the gravity, we can calculate the distance as:

D (Distance) (from A’B’)= r+h 

Therefore, 

I = ∑m (r + h)2

I = ∑m (r2 + h2 + 2rh)

I = ∑mr2 + ∑mh2 + ∑2rh

I = Ic + h2∑m + 2h∑mr

I = Ic + Mh2 + 0

I = Ic + Mh2

Hence, the formula of the theorem of the parallel axis can be derived using the above method. We can now establish the theorem for the parallel axis theorem for a rod. 

For this, the initial step is to calculate and find the moment of inertia of the rod.

The moment of inertia of a rod is given as:

l= ⅓ ML2 

The distance between the rod and the axis is given as:

h= L/2

With this, we arrive at the theorem for the parallel axis of a rod:

lc= ⅓ ML2 -ML2

lc= ⅓ ML2 – ¼ ML2

l= (1/12) ML2  

This is the method to calculate the parallel axis of a rod. The formula of the theorem of the parallel axis is applied to the moment of inertia of a rod, which gives the theorem of the parallel axis for a rod. The next section explains the use of the parallel axis theorem. 

Use of the Parallel Axis Theorem

The use of the parallel axis theorem is varied. These uses cover activities in practical life and other complex fields. The use of the parallel axis theorem has been described below:

• The parallel axis theorem is used to calculate and measure the inertia of a rigid body. 

• The use of the parallel axis theorem can be used to find the moment of inertia of rotational objects in motion. 

• The parallel and the perpendicular axis theorems are related to various areas in engineering. 

• The parallel axis theorem is useful to find the moment of inertia of an object around the axis of the mass of the object. 

• The parallel axis theorem is used to find the centroidal moment of inertia, given the shape of an object is known. 

These are the uses of the parallel axis theorem. In general, the parallel and the perpendicular axis theorems are used to calculate the moment of inertia when an object is rigid. It also calculates the moment of inertia when an object is close or away from the mass.

Conclusion 

This article briefly covers the concept of the moment of inertia. It also explains the parallel axis theorem. The derivation of the formula of the theorem of the parallel axis has also been discussed. The formula for the moment of inertia of a rod has been covered in the article above. Lastly, this article discussed the use of the parallel axis theorem.