What is Torque and Moment?
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.
FAQs on Relation Between Torque and Moment of Inertia
1. What is the fundamental relationship between torque, moment of inertia, and angular acceleration?
The fundamental relationship is expressed by the equation τ = Iα. This is the rotational analogue of Newton's second law of motion (F = ma). In this formula:
- τ (tau) represents the net torque applied to the object, which is the rotational equivalent of force.
- I represents the moment of inertia, which measures the object's resistance to changes in its rotational motion.
- α (alpha) is the angular acceleration, which is the rate at which the object's angular velocity changes.
Essentially, the equation states that the angular acceleration produced is directly proportional to the applied torque and inversely proportional to the moment of inertia.
2. How can the relationship between torque and moment of inertia be understood through an analogy with linear motion?
The relationship τ = Iα can be easily understood by comparing it to Newton's second law for linear motion, F = ma. The terms correspond as follows:
- Torque (τ) is the rotational analogue of Force (F). It's the 'twist' or 'turn' that causes rotation.
- Moment of Inertia (I) is the rotational analogue of mass (m). It represents an object's inherent resistance to being rotated, just as mass represents resistance to being moved linearly.
- Angular Acceleration (α) is the rotational analogue of linear acceleration (a). It's the rate of change of rotational speed.
Just as a larger force is needed to accelerate a heavier object, a larger torque is needed to produce the same angular acceleration in an object with a greater moment of inertia.
3. Is torque directly proportional to the moment of inertia?
The relationship depends on which variable is held constant. For a given angular acceleration (α), the required torque (τ) is directly proportional to the moment of inertia (I). This means if you want to rotate two different objects at the same rate of angular acceleration, the object with the larger moment of inertia will require a greater torque. Conversely, for a given applied torque (τ), the resulting angular acceleration (α) is inversely proportional to the moment of inertia (I).
4. Why is torque considered a vector quantity, and how is its direction determined?
Torque is a vector quantity because it has both magnitude and direction. It is defined as the cross product of the position vector (r) from the axis of rotation to the point of force application, and the force vector (F). The direction of the torque vector is crucial as it indicates the axis and direction of the resulting rotation. This direction is determined by the right-hand rule. If you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the torque vector.
5. What is a real-world example of the relation between torque and moment of inertia?
A classic example is pushing a playground merry-go-round. The torque (τ) is the push you apply. The moment of inertia (I) depends on the mass of the merry-go-round and how that mass is distributed. To make it spin faster (achieve angular acceleration α), you can either push harder (increase force) or push farther from the center (increase the lever arm), both of which increase the torque. If a heavier merry-go-round (larger I) is used, you would need to apply significantly more torque to achieve the same angular acceleration as a lighter one.
6. How does the relationship τ = Iα connect to the principle of conservation of angular momentum?
The more fundamental definition of torque is that it is the rate of change of angular momentum (L), expressed as τ = dL/dt. The formula τ = Iα is a special case derived from this, valid when the moment of inertia (I) is constant. If the net external torque (τ) on a system is zero, then dL/dt = 0. This implies that the total angular momentum (L) of the system must remain constant. This is the principle of conservation of angular momentum. Therefore, the absence of a net external torque is the condition required for angular momentum to be conserved.
7. If a net torque is applied to a body, does its angular velocity always increase?
Not necessarily. A net torque causes angular acceleration, which is the rate of change of angular velocity. If the net torque is applied in the same direction as the object's rotation, its angular velocity will increase. However, if the net torque opposes the direction of rotation (for example, a frictional torque), it will cause a negative angular acceleration (deceleration), and the object's angular velocity will decrease. If the object starts from rest, any net torque will cause its angular velocity to increase from zero.
