Statement of Parallel and Perpendicular Axes Theorems

Introduction

Consistent with the perpendicular axis theorem for an axis perpendicular to the plane, the moment of inertia is equal to the sum of the moments of inertia about two mutually perpendicular axes in its plane.

Ia = Ib+Ic     (Ia = moment of inertia about the given axis perpendicular to the plane of the object, Ib  and Ic are the moments of inertia about two axes mutually perpendicular and running along the plane of the object, also the three axes intersect on the plane of the object.)

Consistent with the parallel-axis theorem, the moment of inertia about any axis equals the sum of the moment of inertia about the axis passing through the centre of mass and parallel to the given axis and the product of mass and square of the distance between the two axes.

I = Ic+Mh2      ( I= moment of inertia about the required axis, Ic = moment of inertia about an axis passing through the centre of mass and parallel to the given axis, M = mass of object,

 h =  distance between the two axes.)

 Consistent with the parallel axis theorem, the moment of inertia of a rigid body about an axis YY is equal to the moment of inertia about another axis Y’Y’ passing through the centre-of-mass G of the body with a direction parallel to YY, plus the product of total mass M of the body and square of the perpendicular distance between the two parallel axes.

The Theorem of Parallel Axis

The theorem of parallel axis states that the moment of inertia of a body about an axis passing via the centre of mass is like the sum of the moment of inertia of the body passing via the centre of mass and the product of mass and square of the gap between them. 

The Theorem of Perpendicular Axis

Perpendicular axis theorem states that for any plane body, the moment of inertia about any of its axes perpendicular to the plane is adequate to the sum of the moment of inertia about any two perpendicular axes within the plane of the body which intersect the primary axis within the plane. 

The perpendicular axis theorem comes into play when the shape of the body is symmetric and about two out of the three axes.

This theorem is valid only for planar objects.

The Formula for the Parallel Axis Theorem

If an object is at rest, it will continue to be at rest unless a force acts on it. Similarly, if an object is in motion, it will continue to be in motion until a force is enacted on it.

If these states of movement or rest of any given object are changed, resistance is encountered. This resistance is understood as inertia. Inertia can alter the speed and direction of motion of the body.

The parallel axis theorem may be expressed in a formula as:

I = Ic + Mh2 

Where: I is the moment of inertia of the body about the required axis, Ic is the moment of inertia about the centre, while M is the mass of the body, and h2 stands for the square of the space between the two axes.

Derivation of the Parallel Axis Theorem

Let us take Ic to be the moment of inertia of an axis passing through the centre of mass. Let I be the moment of inertia about the axis at a distance of h.

Considering a particle of mass m at a distance r from the centre of gravity of the body, Distance from A’B’ = r + h is:

I = ∑m (r + h)2

I is equal to ∑m (r2 + h2 + 2rh)

I = ∑mr2 + ∑mh2 + ∑2mrh

I = Ic + h2∑m + 2h∑mr

I = Ic + Mh2 + 0   ( As about the centre of mass, ∑mr = 0 )

I = Ic + Mh2

This is for the formula for the theorem of parallel axis.

Parallel Axis Theorem of Rod

The moment of inertia gives minimal value when the rotation axis passes through the centre of mass. On the other hand, it increases because the rotation axis is moved beyond the centre of mass.

I = Ic + Mh2

Here, M is the mass of the rod, h is the distance from the centre-of-mass to the parallel axis of rotation, and Ic stands for the moment of inertia at the middle of the mass parallel to the present axis. The moment of inertia Ic for a consistent rod of length L and mass M rotating about an axis through the middle, perpendicular to the rod may be calculated by:

I = Ic + Mh2 → Ic = I – Mh2

Now the moment of inertia about the top of the rod is: I = ML2/3.

The distance from the top of the rod to the middle is h = L/2. Therefore:

Ic = 1/3ML2 – ML2/4

Ic= 1/3ML2 – 1/4ML2

Ic=1/12ML2

The Formula for the Perpendicular Axis Theorem 

The perpendicular axis theorem can be used when the body is in symmetric shape on two out of the three axes. If the moment of inertia of two of the axes is known as the moment of inertia about the third axis, it can be represented in a formula as:

Ia = Ib+Ic   

Example of Applications

Let us assume that in an engineering application, we must find the moment of inertia of a body. However, the body is irregularly shaped. Thus, the moment of inertia can be calculated using the parallel axis theorem to induce the moment of inertia at the centre of gravity of the body. 

This is a very useful theorem in space physics, especially where calculating the moment of inertia of space-crafts and satellites is essential.

The theory of the perpendicular axis, on the other hand, is instrumental in applications where we do not have access to at least one axis of a body, and the moment of inertia of the body must be calculated. This is often applied only to a plane lamina. This theorem is most useful when considering a body is regularly shaped and symmetrical on two out of the three axes. 

Conclusion

The theorems of parallel and perpendicular axes are crucial for calculating the moment of inertia of irregularly shaped objects. These theorems have broad applications in planar dynamics and help solve significant problems in multiple contexts.