Relation Between Torque and Moment of Inertia

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 Newton’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.

In physics and mechanics, torque is rotational but equivalent to a linear force. It is also referred to as the moment, moment of force, rotational force, or turning effect, depending on the field or category of study. The concept originated with the studies performed by Archimedes on the usage of levers. Torque is the twisting force that tends to cause an effective rotation process in any mechanism. The point where the particular object rotates is known as the axis of rotation. Mathematically, torque can be written as T = F * r * sin(theta), and it has units of Newton metres.

Relation Between Inertia and Torque

According to Newton’s first law of motion, the body remains at rest or in the state of motion unless it is driven by an external force. For example, the condenser of the AC, and 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, each particle in the system having their individual rotational masses starts rotating about their axis of rotation. So, this is how we can understand the relationship between the torque and inertia by applying Newton’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.

Concept Used in Rotational Motion

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.

Derive Relation Between Torque and Moment of Inertia

Let’s suppose that a particle ‘Q’ of mass ‘m’ is rotating around the axis of rotation where it is making an arc along the circle of radius ‘r’. Now, according to Newton’s second law of motion, we have:

F = ma

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

Now,\[a=\frac{F}{m}\]…….(a)

If we say that the particle is moving along the circle with displacement ‘s’, then we can rewrite the equation for linear acceleration as the double derivative of angular displacement in the following manner:

\[a=\frac{d}{dt}\frac{d(s)}{dt}\]……..(1)

Since a system has an ‘n’ 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: ‘s = r’, let’s apply this in the equation (1):

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

Also, we know the relation between the linear and the angular acceleration, for which let’s write the same:

\[a=r\alpha \]……(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, let’s see how it happens:

\[r\frac{d\theta }{dt}=\omega \]

And,

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

Here, is the angular acceleration of particle ‘Q’. So, we derived equation (3) as well. Now, proceeding to the next step:

Force and Moment of Force

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

\[\tau =rF\]

\[F=\frac{\tau }{r}\]…..(4)

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

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

Adjusting the above equation, we get:

\[m\alpha r^2=\tau \]…..(5)

We also know that \[I=mr^2\]….(6)

Where

m = mass of the particle ‘Q’ 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 equation (6) in (5), we get:

\[\tau =I\alpha \]…..(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:

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

We call this equation the fundamental law of rotational motion or the law of rotational motion. Now, let’s define the above equation:

Definition of Fundamental Law of Rotational Motion

If α = 1, then τ = 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 an ‘n’ number of particles, and each particle follows equation (7), so the law of rotational motion applies to each and every particle of the system.