Torque

Consider attempting to open a door by pushing close to the hinges versus pushing at the edge. Which is simpler? The latter is simpler due to torque, a basic physics and engineering concept. 

Torque is important in machines, cars, and even everyday life. In this article, we will learn about what torque is, its mathematical definition, everyday applications, and misconceptions.

What is a Torque?

Torque is the measure of the rotational force acting on an object around a specific axis. It indicates how effectively a force can make an object rotate. Essentially, torque is the "twisting force" that causes an object to turn, spin, or rotate. In the context of rotational motion, torque plays a role similar to that of force in linear motion — it causes angular acceleration rather than linear acceleration. 

Because of this, torque is often described as the rotational equivalent of linear force. The axis of rotation is the straight line around which the object rotates. In physics, torque is understood as the force’s ability to induce twisting or turning, and it is also commonly referred to as the moment or moment of force. The distance between the axis of rotation and the point of force application is called the moment arm or lever arm.

Torque Definition

Torque, or the moment of force, is a measure of the force that causes rotation of an object. It specifies how efficiently a force can make an object turn around an axis.

Mathematically, torque (τ) is given by: τ = r × Fsin(θ) where:

• r = Perpendicular distance from the axis of rotation (lever arm)
• F = Applied force
• θ = Angle between force and lever arm

Torque Definition

Torque, or the moment of force, is a measure of the force that causes rotation of an object. It specifies how efficiently a force can make an object turn around an axis.

Mathematically, torque (τ) is given by: τ = r × Fsin(θ) where:

• r = Perpendicular distance from the axis of rotation (lever arm)
• F = Applied force
• θ = Angle between force and lever arm

Torque Formula

Torque is defined as the rotational equivalent of force. It is the force that causes an object to rotate around an axis.

The formula for torque (τ) is:

τ = r · F · sin(θ)

Where:

• τ is the torque
• r is the distance from the pivot point (axis of rotation) to the point where the force is applied
• F is the magnitude of the applied force
• θ is the angle between the force vector and the lever arm (the vector from the axis of rotation to the point where the force is applied)

If the force is applied perpendicular to the lever arm, sin(θ) becomes 1, and the formula simplifies to:

τ = r · F

Torque Dimensional Formula

The dimensional formula of torque can be derived from the formula for torque.

Given:

• Force F has the dimensional formula: [M L T^-2]
• Distance r has the dimensional formula: [L]

So, the dimensional formula for torque τ is:

[τ] = [r] · [F] = [L] · [M L T^-2] = [M L² T^-2]

Thus, the dimensional formula for torque is: [M L² T^-2]

Derivation of Torque

Torque is a measure of the rotational force applied to an object around an axis. To derive the formula:

Consider a force F applied to an object at a distance r from the axis of rotation, making an angle θ with the lever arm (line joining the axis of rotation to the point of application of force).

The component of the force responsible for causing the rotational effect is F sin(θ).

The rotational force (or torque) is the force applied at a distance r from the axis, hence:

τ = r · F · sin(θ)

This is the general form of the torque equation. If the angle θ is 90°, the force is fully effective in causing rotation, and the formula simplifies to τ = r · F.

Torque Units

The SI unit of torque is the Newton-meter (Nm).

• Newton (N) is the unit of force.
• Meter (m) is the unit of distance.

So, the unit of torque is the product of these two, i.e., Newton-meter (Nm).

In terms of fundamental SI units, it can be written as:

[τ] = kg · m² · s^-2

How is Torque Calculated?

Torque is a measure of the rotational force applied to an object about an axis. It is calculated using the following general formula:

τ = r · F · sin(θ)

Where:

• τ is the torque (measured in Newton-meters, Nm)
• r is the distance from the axis of rotation to the point where the force is applied (measured in meters, m)
• F is the magnitude of the applied force (measured in Newtons, N)
• θ is the angle between the applied force vector and the lever arm (measured in degrees or radians)

Steps to Calculate Torque

1. Identify the distance (r): Measure the distance from the axis of rotation to the point where the force is applied.
2. Determine the force (F): Identify the magnitude of the applied force.
3. Calculate the angle (θ): Measure the angle between the applied force and the lever arm. If the force is applied perpendicular to the lever arm, θ will be 90°, and sin(90°) = 1, simplifying the calculation.
4. Substitute the values into the formula: Use the formula τ = r · F · sin(θ) to calculate the torque. If the angle is 90° (force is perpendicular to the lever arm), the formula simplifies to τ = r · F.

Applications of Torque

Torque plays a significant role in many areas of physics and engineering. Here are some of the primary applications of torque in everyday life and engineering systems:

• Rotational Motion in Machinery
• Wrenches and Tools
• Vehicle Engines and Rotating Parts
• Levers and Pulleys
• Bicycle and Sports Equipment
• Electric Motors and Generators
• Torsional Vibration Analysis
• Human Body and Joints
• Wind Turbines
• Gyroscopes

Solved Examples

Q. A person applies a force of 20 N at the edge of a door, 0.8 m from the hinges, at an angle of 90°.

Ans. Using the torque formula: τ = r × Fsin(θ)

Substituting values:

τ = (0.8 m) × (20 N) × sin(90°)

Since sin(90°) = 1, we get: τ = 0.8 × 20 = 16 Nm

Q. A mechanic uses a 0.5 m long wrench to apply a force of 50 N at an angle of 60° to the handle. What is the torque produced?

Ans. τ = r × Fsin(θ)

Substituting values:

τ = (0.5 m) × (50 N) × sin(60°)

Using sin(60°) ≈ 0.866, we get:

τ = 0.5 × 50 × 0.866 = 21.65 Nm