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We know that the turning effect of the fan is called the torque and there is a direct relationship between the torque and the moment of inertia.

So, here on switching the button of the fan, when the fan begins to rotate, its moment of force or torque varies inversely with the acceleration. Here, this relationship can be considered as the sister of Newtonâ€™s second law of motion. The moment of inertia is the rotational mass of the fan and the torque is its rotational force or its turning force.

According to Newtonâ€™s first law of motion, the body remains at rest or in the state of motion unless it is driven by an external force. So, here when the condenser of the AC, washing machine remains at rest unless we switch on the power button, and allows it to rotate with the help of electricity.

So, we can see that all the rotating electrical appliances remain at rest, and when the turning effect or the torque is offered, where each particle in the system having their individual rotational masses start rotating about their axis of rotation. So, this is how we can understand the relationship between the torque and inertia by applying Newtonâ€™s first law of motion.

So far we understood the relationship of torque with inertia and moment of inertia. Now, we will derive the relation between torque and moment of inertia.

According to the mechanics of rotational motion, every rigid body executing rotational motion about the fixed axis bears a uniform angular acceleration motion, i.e., under the action of torque or the moment of force.

Letâ€™s suppose that a particle â€˜Qâ€™ of mass â€˜mâ€™ is rotating around the axis of rotation where it is making an arc along the circle of radius â€˜râ€™. Now, according to Newtonâ€™s second law of motion, we have:

F = ma

Where a is the acceleration by which the body is rotating.

Now, a = F/mâ€¦.(a)

If we say that the particle is moving along the circle with displacement â€˜sâ€™, then we can rewrite the equation for linear acceleration as the double derivative of angular displacement in the following manner:

\[a = \frac{d}{dt} \frac{d(s)}{dt}\] â€¦.(1)

Since a system has â€˜nâ€™ number of particles, so the acceleration of each particle will be a1, a2, a3,...., an.

For a body executing the rotational motion, the relation is: â€˜s = râ€™, letâ€™s apply this is in the equation (1):

\[a = \frac{d}{dt} \frac{d(r \theta)}{dt}\]

Since â€˜râ€™ is the radius of the circle along which the particle is rotating and it is a constant value, so taking it out of the derivative form and rewriting the equation in the following manner:

\[a = \frac{d}{dt} \frac{rd(\theta)}{dt}\] â€¦.(2)

Also, we know that the relation between the linear and the angular acceleration, for which letâ€™s write the same:

\[aÂ = r \alpha\]â€¦â€¦(3)

From, equation (2), we are getting equation (3), as we can see that the rate of change of angular displacement is the angular velocity, letâ€™s see how it happens:

\[\frac{rd(\theta)}{dt} = \omega\]

And,

\[\frac{d \omega}{dt} = \alpha\]

Here, is the angular acceleration of particle â€˜Qâ€™. So, we derived equation (3) as well. Now, proceeding to the next step:

We know one more relationship and that is between the force applied to the body and the torque, and that is as follows:

\[\tau = rF\]

\[F = \frac{\tau}{r}\] â€¦..(4)

Substituting the value of equation (3) and (4) in equation (a) we get:

\[r \alpha = \frac{(\frac{\tau}{r})}{m}\]

Adjusting the above equation, we get:

\[\alpha m r^{2} = \tau\] â€¦..(5)

We also know that \[I = mr^{2}\] â€¦.(6)

Where,

m = mass of the particle â€˜Qâ€™ and r is the square of the distance of the particle from the axis of rotation or simply the radius of gyration, so substituting the value of eq (6) in (5), we get:

\[\tau = I \alpha\] â€¦..(7)

So, equation (7) is the desired equation for which we have done all this mathematical derivation. Equation (7) describes the ultimate relationship between moment of inertia and torque.

We can rewrite the equation (7) in the vector form as:

\[\overrightarrow{\tau} = I \overrightarrow{\alpha}\]

We call this equation the fundamental law of rotational motion or the law of rotational motion. Now, letâ€™s define the above equation:

If Î± = 1, then Ï„ = I * 1. From this statement, we can say that the moment of inertia and the torque applied to the body are equal to each other in the absence of angular acceleration.

Since the system has â€˜nâ€™ number of particles, and each particle follows equation (7), so the law of rotational motion applies to each and every particle of the system.

FAQ (Frequently Asked Questions)

Question 1: What are the Significance of Torque and the Moment of Inertia?

Answer: The torque is similar to the applied force in the linear motion. It is a fundamental criterion that keeps the body in rotational motion. So when the torque is offered to the body, it starts making rotations with uniform angular acceleration.

Question 2: Name Two Theorems on the Moment of Inertia.

Answer: When we are known with the moment of inertia of an object or a system about a given axis, then we can calculate the moment of inertia about any other axis easily. This can be done by the following two theorems:

Parallel axes theorem.

Perpendicular axes theorem.

Question 3: Is Torque a Vector Quantity?

Answer: Yes, the torque is a vector quantity. It is equivalent to the vector product of the vector pointing from the axis to the point of application of force applied and the vector of force. The torque points upward from a counterclockwise rotation, and vice-versa.

Question 4: Is the Moment of Inertia an Extensive Property?

Answer: Yes, the moment of inertia is an extension (additive) attribute for a point mass object.

We define the moment of inertia as the product of the mass and the square of the perpendicular distance to the axis of rotation.

For a rigid system, it is the sum of the moments of inertia of its component subsystems when taken about the same axis of rotation.