Theorems of Parallel and Perpendicular Axis

Suppose you have an object like a ball or a rotating disc that rotates about its centre. Now, if you move the point of rotation to the point other than its centre. How do you find the moment of inertia? The answer to this question is by the parallel axis theorem.

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Before we study the theorems of parallel and perpendicular axis let us first see what moment of inertia is. Moment of inertia is the property of the body due to which it resists angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation.

 

∴ Moment of inertia I = Σ m_ir_i²

So in order to calculate the moment of inertia we use two important theorems which are the perpendicular and parallel axis theorem.

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• Moment of Inertia
• Rolling Motion
• Angular Velocity and Angular Acceleration
• Linear Momentum of System of Particles
• Torque and Angular Momentum
• Equilibrium of a Rigid Body
• Angular Momentum in Case of Rotation About a Fixed Axis
• Dynamics of Rotational Motion About a Fixed Axis
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Perpendicular Axis Theorem                                                                                                                     

This theorem is applicable only to the planar bodies. Bodies which are flat with very less or negligible thickness. This theorem states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.

( Source: Toproadrunner5 )

In the above figure, we can see the perpendicular body. So Z axis is the axis which is perpendicular to the plane of the body and the other two axes lie in the plane of the body. So this theorem states that

I_{Z =}  I_x  +  I_y

That means the moment of inertia about an axis which is perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes.

Let us see an example of this theorem:

Suppose we want to calculate the moment of inertia of a uniform ring about its diameter. Let its centre be MR²/2, where M is the mass and R is the radius. So, by the theorem of perpendicular axes, I_{Z =}  I_x  +  I_y. Since the ring is uniform, all the diameters are equal.

∴  I_{x  =} I_y

∴ I_Z  = 2 I_x

I_z = MR²/4

So finally the moment of inertia of a disc about any of its diameter is MR²/4

Learn more about Moment of Inertia in detail here.

Parallel Axis Theorem

Parallel axis theorem is applicable to bodies of any shape. The theorem of parallel axis states that the moment of inertia of a body about an axis parallel to an axis passing through the centre of mass is equal to the sum of the moment of inertia of body about an axis passing through centre of mass and product of mass and square of the distance between the two axes.

I_Z’ = I_z + Mα²

where, α is the distance between two axes.

Solved Examples For You

Q1. The moment of inertia of a thin uniform rod of mass M and length L bout an axis perpendicular to the rod, through its centre is I. The moment of inertia of the rod about an axis perpendicular to the rod through its endpoint is:

1. I/4
2. I/2
3. 2I
4. 4I

Answer: D. I_centre =  ML²/12 and I_endpoint = ML²/3 = 4I

Q2. The radius of gyration of a body is 18 cm when it is rotating about an axis passing through a centre of mass of a body. If the radius of gyration of the same body is 30 cm about a parallel axis to the first axis then, the perpendicular distance between two parallel axes is: