Dimensional Formula of Torque – Complete Theory, Derivation and Detailed Solution

Correct Answer

2

Understanding the Concept of Dimensions

Every physical quantity can be expressed in terms of a few fundamental quantities called base quantities. In mechanics, the three most important base quantities are mass (M), length (L), and time (T).

The dimensional formula of any derived quantity tells us how that quantity depends on these fundamental quantities. It is extremely useful for checking the correctness of equations, deriving relationships between variables, converting units, and solving numerical problems efficiently.

What is Torque?

Torque, also known as the moment of force, is the turning effect produced by a force acting at some distance from a fixed axis or pivot. It determines how effectively a force can rotate an object.

Examples of torque can be seen while opening a door, tightening a bolt with a wrench, turning a steering wheel, or using a screwdriver. In each case, applying the force farther from the axis of rotation produces a larger turning effect.

Formula for Torque

The magnitude of torque is given by

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Torque = Force × Perpendicular Distance

or

τ = F × r

where

• τ = Torque
• F = Applied force
• r = Perpendicular distance from the axis of rotation

Step 1: Write the Dimensional Formula of Force

From Newton’s Second Law,

Force = Mass × Acceleration

The dimensional formula of acceleration is

[Acceleration] = LT⁻²

Therefore,

[Force] = M × LT⁻²

[Force] = MLT⁻²

Step 2: Determine the Dimensional Formula of Torque

Since torque is the product of force and distance,

[Torque] = [Force] × [Distance]

Substituting the dimensional formula of force,

[Torque] = (MLT⁻²) × L

[Torque] = ML²T⁻²

Step 3: Identify the Exponents

Comparing

ML²T⁻²

with

M^xL^yT^z,

we obtain

• x = 1
• y = 2
• z = −2

Therefore,

x + y + z = 1 + 2 − 2 = 1

Detailed Explanation

Torque depends on both the magnitude of the applied force and the perpendicular distance from the axis of rotation. Since force itself depends on mass, length, and time, multiplying it by another length increases the power of length by one while leaving the powers of mass and time unchanged.

Although torque and work have the same dimensional formula, they represent entirely different physical quantities. Work is a scalar quantity that measures energy transfer, whereas torque is a vector quantity that measures the tendency of a force to produce rotation. Their SI units are both expressed as newton metre (N·m), but they should never be used interchangeably.

Why Dimensional Analysis is Important

Dimensional analysis helps physicists verify equations before performing experiments. If the dimensions on the left-hand side of an equation do not match those on the right-hand side, the equation cannot be physically correct. This technique is widely used in mechanics, thermodynamics, electromagnetism, fluid mechanics, and quantum physics.

Real-Life Applications of Torque

Torque plays an important role in everyday life and engineering. Mechanics use long-handled spanners because increasing the distance from the pivot increases torque, making it easier to loosen tight bolts. Automobile engines generate torque to rotate the wheels, electric motors convert electrical energy into rotational motion using torque, and turbines in hydroelectric and wind power plants operate by producing rotational torque.

Final Answer

The dimensional formula of torque is

ML²T⁻²

Hence,

x = 1, y = 2, z = −2

Therefore,

x + y + z = 1

Final Answer: 1