Relation between Moment of Inertia and Torque: Derivation

Relation between Moment of Inertia and Torque can be established with the help of Newton’s Second Law of Motion. Torque is a unit of measurement for the force required to rotate an object around an axis. Whereas, the rotational analogue of mass for linear motion, is known as the moment of inertia. When you turn on the fan and it starts to whirl, the fan's moment of force or torque varies inversely with the acceleration. The fan's rotating mass is its moment of inertia, and the torque is its rotational force or turning force.

Key Takeaways: Moment of Inertia, Torque, Rotational Motion, Angular Acceleration, SI unit, Rotation Axis

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What is Moment of Inertia?

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• Moment of Inertia is the sum of the product of the mass of each particle with its square of the distance from the axis of rotation.
• It is the quantity denoted by the body resisting angular acceleration.

• To put it another way, it's a number that specifies how much torque is needed for a particular angular acceleration in a rotating axis.
• The moment of inertia is also known as angular mass or rotational inertia.
• Kilogram square meter (kgm²) is the SI unit for moment of inertia.

The moment of inertia is commonly expressed in terms of a rotational axis. It is mostly determined by the distribution of mass around a rotational axis. It varies based on which axis is selected.

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Formula of Moment of Inertia

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Moment of Inertia is giving by,

I = m x r²

Where,

I → ∫dI = ∫₀^M r² dm

M → Sum of the mass's products

r → Distance from the rotation's axis

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What is Torque?

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• Torque is the force that causes an object to twist along an axis.
• In linear kinematics, force causes an object to accelerate.
• Torque is also responsible for angular acceleration.
• As a result, torque can be defined as the linear force's rotational equivalent. 
• The axis of rotation is the point at which the item rotates.
• Torque is described using a variety of terminologies, including moment and moment of force.
• S.I unit of torque is newton-meter (Nm).

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Formula of Torque

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Formula of torque is,

τ = F.r. sinθ

Where,

τ →  Torque

F → Force vector

r → Position vector

θ → The angle formed by the force vector and the lever arm vector

Torque can be either static or dynamic. Static torque is torque that does not induce an angular acceleration. Consider the following examples of torque:

• When a person pushes a closed door, the door remains static since it does not rotate despite the force applied. Similarly, pedaling a bicycle at a constant speed is an example of static torque because there is no acceleration.

• Because the car is moving along the track, the drive shaft in a racing car speeding from the start line must be creating an angular acceleration of the wheels, resulting in dynamic torque.

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Relation between the Moment of Inertia & Torque

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Newton's first law of motion states that unless a body is propelled by an external force, it will remain at rest or in motion. So, unless we turn on the power button and allow the washing machine to revolve with the help of electricity, it remains at rest.

As a result, all rotating electrical appliances remain at rest until the turning effect or torque is applied, at which each particle in the system with its rotational mass begins to rotate around its axis of rotation.

So, by utilizing Newton's first law of motion, we may comprehend the link between torque and inertia. Every rigid body executing rotational motion along a fixed axis bears a uniform angular acceleration motion, i.e., under the action of torque or the moment of force, according to rotational motion mechanics.

Consider an entity in rotatory motion with mass m traveling along an arc of a circle with radius r. We know from Newton's Second Law of Motion that,

F= ma

→ a= f/m -------- (1)

As we know, 

\(\alpha = \frac {d} {dt} (\frac{ds} {dt})\)

The relationship for a body doing a rotational motion is ‘s = r'.

Therefore, 

\(\frac {d} {dt} (\frac {rd\theta} {dt}) = r \frac {d} {dt} (\frac {d\theta} {dt})\)

As a result, 

a = rα is the angular acceleration------- (2)

Similarly, Torque should be substituted for Force F

τ = Fr 

F = τ / r ------- (3)

By substituting eq (2) and (3) in (1) we get:

τ = m r² α

We know that, Moment of inertia is I= mr²

Therefore, 

τ = I α

Where 

α → Angular acceleration

The above equation is called the fundamental law of rotational motion, or the law of rotational motion.

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Things To Remember

• Relationship between Moment Of Inertia and Torque can be established with the help of Newton’s Second Law of Motion.
• Torque is a unit of measurement for the force required to rotate an object around an axis.
• The sum of the product of the mass of each particle with its square of the distance from the axis of rotation is called moment of inertia.
• S.I unit of torque is newton-meter (Nm).
• Kilogram square meter (kgm²) is the SI unit for moment of inertia.
• The relationship between Torque and moment of inertia is given by the equation τ = I α​

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