Perpendicular Axis Theorem

The parallel and perpendicular axis theorem deals with the moment of inertia. So, before studying the theorems, let's know about the moment of inertia. It is the property of a body by virtue of which a body resists angular acceleration. 

Angular acceleration is the sum of the product of masses of particles of the body with the square of the distance from the axis of rotation.

Moment of inertia \[I_{i} = \sum m_{i}r_{i}^{2}\]

The equation is applicable only in-plane lamina. 

This theorem is used for symmetric objects, i.e. when two out of three axes are symmetric. The moment of inertia of the third axis can be calculated by this equation when the moment of inertia of the other two axes is known.

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State and Prove Theorem of Perpendicular Axis

Perpendicular axis theorem statement - The perpendicular axis theorem states that the moment of inertia, for any axis which is perpendicular to the plane, is equal to the sum of any two perpendicular axes of the body which intersects with the first axis. 

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Let us consider a plane lamina made up of a large number of particles in the x-y plane as represented by the figure. Consider a particle having mass 'm' at point P. 

From P, draw PN and PN' perpendicular to the x and y-axis, respectively.

The moment of inertia about the x-axis = my².

The moment of inertia of the whole lamina about the x-axis is given by 

Ix = ∑my²-----(1)

The moment of inertia of the whole lamina about why the axis is given by

Iy = ∑mx²-------(2)

Similarly, the moment of inertia of the whole lamina about the z-axis is given by,

Iz = ∑mr²

But r² = x² + y²

Therefore,

Iz = ∑m (x² + y²)

From eq(1) and (2), we get:

i.e., Iz = ∑mx² + ∑my²

      (or)

Iz = Ix + Iy.

The perpendicular axis theorem helps calculate the moment of inertia of a body where it's difficult to access one vital axis of the body. 

Parallel Axis Theorem and Perpendicular Axis Theorem

The parallel and perpendicular axis theorem is mentioned below:

1. Parallel Axis Theorem

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

Statement

The parallel axis theorem is represented by the following equation:

I = Ic + Mh2

Where,

I = moment of inertia of the body.

Ic = moment of inertia about the centre of the body.

M = mass of the body.

h2 = square of the distance between the two axes.

2. Perpendicular Axis Theorem

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Let’s see an example for this theorem:

Let us suppose that we want to calculate the moment of inertia of a uniform ring about its diameter.

Let the centre of the ring be MR2/2,

Where,

M = mass of the ring

R = radius of the ring

According to the perpendicular axis theorem,

Iz = Ix + Iy

Since it is a uniform ring; so all its diameter is equal

Therefore, \[I_{x} = I_{y}\]  \[I_{z} = 2 I_{x}\]

Ix  = Iy    

∴ Iz = 2 Ix

Iz = (MR²)/2

Hence, the moment of inertia of a ring about its diameter is MR2/2.

Parallel Axis Theorem Derivation

Let Ic be the moment of inertia of an object with the axis passing through the centre of mass of the object. Let, I will be the moment of inertia about the axis A'B' (from figure AB) at a distance of h.

Let us consider a particle having mass ‘m’ and located at a distance ‘r’ from the centre of gravity of the body.

Distance from A’B’ = r + h

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

Difference Between Parallel and Perpendicular Axis Theorem