Calculating Moment Of Inertia

Establishing the location of the X, Y, and Z axes is the first step in calculating the moment of inertia for a mass. How carefully the axes are defined will have a significant impact on the accuracy of the calculations (and the measurements used to validate the calculations). These axes can theoretically be located anywhere about the object under consideration, as long as they are mutually perpendicular. However, calculating the moment of inertia is useless unless the axes are correctly defined and can be accurately referenced.

Inertia is comparable to the moment of inertia; however, it pertains to rotation instead of linear motion. The tendency of an object to stay at rest or continue travelling in a straight path at the same speed is known as inertia. Mass can be thought of as a synonym for inertia. As a result, rotational mass is defined as the moment of inertia. Unlike inertia, MOI is also affected by the mass distribution in an object. The greater the mass’s distance from the rotation’s centre, the larger the moment of inertia.

For rotation, formulas for calculating the moment of inertia:

F = Ma (F = force; M = mass; a = linear acceleration)

T = IA (T = Torque = inertia; I = moment of inertia; A= rotational acceleration

Moment of Inertia Example

Talking about the calculating moment of inertia in detail, consider an example:

• Assume you’re currently on a bus. You find a seat and take a seat.
•  The bus begins to go onward. 
• You arrive at a bus stop, and the bus stops. What were your feelings at this point? When the bus stopped, your upper body pushed forward, but your lower body remained stationary.

What is the reason for this? It’s due to the law of inertia. Your lower body is in immediate contact with the bus, but your upper body is not. As a result, as the bus came to a stop, your lower body came to a halt, but your upper body continued to go ahead, resisting change in its condition.

Similarly, a force pushes you back when you board a moving train. This is because you were at rest before boarding the train. Your lower body touches the train as soon as you board it, but your upper body remains stationary. As a result, it is pushed backwards, resisting change in its state.

Moment of Inertia Formulas

Here are the formulas for calculating the moment of inertia:

Solid Sphere

A solid sphere with mass M and radius R revolving on an axis that passes through its centre has a moment of inertia specified by the formula:

I is equal to (2/5)MR2

Hollow Thin-Walled Sphere

With mass M and radius R, a hollow sphere with a thin, insignificant wall revolving on an axis that passes through the centre of the sphere has a moment of inertia specified by the formula:

I is equal to (2/3)MR2

Solid Cylinder

A solid cylinder with mass M and radius R revolving on an axis that passes through its centre has a moment of inertia calculated by the formula:

I is equal to (1/2)MR2

Hollow Thin-Walled Cylinder

With mass M and radius R, a hollow cylinder with a thin, negligible wall revolving on an axis through the centre of the cylinder has a moment of inertia specified by the formula:

I is equal to MR2

Hollow Cylinder

A hollow cylinder with mass M, internal radius R1, and external radius R2 rotates on an axis that runs through the centre of the cylinder and has a moment of inertia specified by the formula:

I is equal to (1/2)M(R12 + R22)

Note: You may find the formula for the moment of inertia of a hollow thin-walled cylinder by setting R1 = R2 = R (or, more precisely, by taking the mathematical limit as R1 and R2 approach a standard radius R).

Rectangular Plate, Axis Through Centre

A thin rectangular plate with mass M and side lengths a and b, spinning on an axis perpendicular to the plate’s centre, has a moment of inertia specified by the formula:

I is equal to (1/12) M (a2 + b2)

Rectangular Plate, Axis Along Edge

The moment of inertia of a thin rectangular plate spinning on an axis along one edge of the plate, (with mass M and side lengths a and b, where an is the distance perpendicular to the axis of revolution) is calculated as follows:

I is equal to (1/3)Ma2

Slender Rod, Axis Through Centre

A slender rod with mass M and length L revolving on an axis that passes through its centre (perpendicular to its length) has a moment of inertia calculated by the formula:

I is equal to (1/12)ML2

Conclusion

The basic conceptual understanding of the formulas for calculating the moment of inertia:

• Any rotating object’s moment of inertia may be calculated by measuring the distance between each particle and the axis of rotation (r in the equation), squaring that value (the r2 term), and multiplying it by the particle’s mass.
• In this article we saw formulas of moment of inertia of different objects.