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Dimensions of physical quantities

NCERT Class 11 Physics, Chapter 1: Units and Measurement, §1.4–1.5. A short visual lesson: 2 narrated 3D videos, 1 interactive simulation and 2 quiz questions.

What are dimensions?

Narrated video. Transcript:

Here’s a powerful idea: every physical quantity, however complicated, is built from a few base quantities.

In mechanics we need only three: mass, length and time. We write their dimensions as M, L and T inside square brackets.

Take volume: length times breadth times height. Watch three L blocks join into a cube. Volume has dimensions [L³].

Now velocity: displacement divided by time. One L on top, one T below. Its dimensions are [L T⁻¹].

Acceleration is velocity divided by time again, so [L T⁻²]. Force is mass times acceleration, so [M L T⁻²].

These powers are called the dimensions of force: 1 in mass, 1 in length and minus 2 in time.

Square brackets show only the kind of quantity, not its size. Initial velocity, final velocity, average velocity and speed all have the same dimensions, [L T⁻¹].

Pure numbers like 2 or π have no dimensions. Neither does an angle, which is arc divided by radius. We write this as [M⁰ L⁰ T⁰].

Dimensional formulae and equations

Narrated video. Transcript:

Writing a quantity in terms of M, L and T gives its dimensional formula. For velocity it is [M⁰ L T⁻¹].

Put an equals sign between the quantity and its formula, and you have a dimensional equation: [v] = [M⁰ L T⁻¹].

Let’s build a few. Density is mass per unit volume: [M] over [L³] gives [M L⁻³].

Work is force times displacement: [M L T⁻²] times [L] gives [M L² T⁻²]. Every kind of energy has these same dimensions.

Power is work per unit time: [M L² T⁻³]. Pressure is force per unit area: [M L⁻¹ T⁻²].

Now look closely. Torque, force times distance, also comes out as [M L² T⁻²]. Two different quantities can share the same dimensions.

The dimensional formula also gives the SI unit. [M L T⁻²] means kilogram metre per second squared: the newton.

Your turn. In the next activity, stack up M, L and T blocks to build any quantity you like.

Try it: build a dimensional formula

Interactive simulation. Things to try:

  • Pick a quantity from the list to see how its formula is built.
  • Positive powers stack up; negative powers (dividing) hang down.
  • Press “Quiz me” and set the sliders to match the quantity named.

Quick check

What is the dimensional formula of pressure?

  1. [M L T⁻²]
  2. [M L⁻¹ T⁻²]
  3. [M L² T⁻²]
  4. [M L⁻² T⁻²]
Answer

B. Pressure = force ÷ area = [M L T⁻²] ÷ [L²] = [M L⁻¹ T⁻²]. The first option is force and the third is energy.

Same dimensions?

Which pair of quantities has the same dimensions?

  1. Work and torque
  2. Force and momentum
  3. Power and energy
  4. Velocity and acceleration
Answer

A. Work = force × displacement and torque = force × perpendicular distance. Both are [M L² T⁻²]. Momentum is [M L T⁻¹], not [M L T⁻²], and power is energy per second, [M L² T⁻³].

Summary

  • Velocity: [v] = [M⁰ L T⁻¹]
  • Force: [F] = [M L T⁻²]
  • Work, energy: [W] = [M L² T⁻²]
  • Pressure: [P] = [M L⁻¹ T⁻²]
  • The dimensions of a quantity are the powers to which the base quantities must be raised to represent it.
  • A dimensional formula such as [M L T⁻²] shows those powers; writing [F] = [M L T⁻²] is a dimensional equation.
  • Different quantities can share dimensions, such as work and torque, or frequency and angular velocity.
  • Pure numbers and angles are dimensionless: [M⁰ L⁰ T⁰].