System of Particles and Rotational Dynamics

Moment of Inertia: Formula, Definition, and Examples

Moment of inertia aka angular mass or rotational inertia can be defined w.r.t. rotation axis, as a quantity that decides the amount of torque required for a desired angular acceleration or a property of a body due to which it resists angular acceleration. The formula for moment of inertia is the “sum of the product of mass” of each particle with the “square of its distance from the axis of the rotation”. The formula of Moment of Inertia is expressed as I = Σ miri2.

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Moment of Inertia Example

Imagine you are on a bus right now. You find a seat and sit down. The bus starts moving forward. After a few minutes, you arrive at a bus stop and the bus stops. What did you experience at this point? Yes. When the bus stopped, your upper body moved forward whereas your lower body did not move.

Why is that? It is because of InertiaYour lower body is in contact with the bus but your upper body is not in contact with the bus directly. Therefore, when the bus stopped, your lower body stopped with the bus but your upper body kept moving forward, that is, it resisted change in its state.

Similarly, when you board a moving train, you experience a force that pushes you backward. That is because before boarding the train you were at rest. As soon as you board the moving train, your lower body comes in contact with the train but your upper body is still at rest. Therefore, it gets pushed backward, that is, it resists change in its state.

Understand the Theorem of Parallel and Perpendicular Axis here in detail.

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What is Inertia?

What is Inertia? It is the property of a body by virtue of which it resists change in its state of rest or motion. But what causes inertia in a body? Let’s find out.

Inertia in a body is due to it mass. More the mass of a body more is the inertia. For instance, it is easier to throw a small stone farther than a heavier one. Because the heavier one has more mass, it resists change more, that is, it has more inertia.

Moment of Inertia Definition

So we have studied that inertia is basically mass. In rotational motion, a body rotates about a fixed axis. Each particle in the body moves in a circle with linear velocity, that is, each particle moves with an angular acceleration. 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.

Formula for Moment of Inertia can be expressed as:

∴ Moment of inertia I = Σ miri2

Kinetic Energy in Rotational Motion

What is the analogue of mass in rotational motion? To answer this question, we have to derive the equation of kinetic energy in rotational motion.

moment of inertia

Consider a particle of mass at a distance from the axis with linear velocity = vi = riω. Therefore, the kinetic energy of this particle is,

ki = 1/2 mivi2 = 1/2  mi(r)2ω2

where mi is the mass of the particle. The total kinetic energy K of the body is thus the sum of the kinetic energies of individual particles. ∴ K = Σ k= 1/2 ( Σ mi(r)2ω2) where is the number of particles in the body. We know that angular acceleration ω is the same for all particles. Therefore, taking ω out of the sum, we get,

K = 1/2ω2 (Σ mi(r)………………….(I)

Let Σ mi(r)2 be I which is a new parameter characterising the rigid body known as the Moment of Inertia.  Therefore, with this definition,

K = 1/2Iω……………..(II)

The parameter I is independent of the magnitude of the angular velocity. It is the characteristic of the rigid body and the axis about which it rotates. We already know that linear velocity in linear motion is analogous to angular acceleration in rotational motion.

On comparing equation II with the formula of the kinetic energy of the whole rotating body in linear motion, it is evident that mass in linear motion is analogous to the moment of inertia in rotational motion. Hence, the question is answered. In simple words, moment of inertia is the measure of the way in which different parts of the body are distributed at different distances from the axis.

Radius of Gyration

As a measure of the way in which the mass of a rotating rigid body is distributed with respect to the axis of rotation, we define a new parameter known as the radius of gyration. It is related to the moment of inertia and the total mass of the body. Notice that we can write I = Mkwhere k has the dimension of length.

Therefore, the radius of gyration is the distance from the axis of a mass point whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis. Therefore, the moment of inertia depends not only on the mass, shape, and size of the body but also the distribution of mass in the body about the axis of rotation.

Learn more about Torque and Angular Momentum here.

Here’s a Solved Question for You

Q: The moment of inertia depends on:

a) mass of the body                                                           b) shape and size of the body

c) distribution of mass about the axis of rotation       d) all of the above

Solution: d) all of the above. All the factors together determine the moment of inertia of a body.

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