Dimensions of Physical Quantities

Chapter 1 — Physical Quantities and Measurements · Topic: Dimensions of Physical Quantities

Learning Outcomes

• Define the dimension of a physical quantity.
• Distinguish between dimensions and units.
• Recall the seven SI base dimensions.
• Write a general dimensional formula.
• Derive the dimensions of common physical quantities.
• Identify dimensionless quantities.
• Determine dimensional exponents.
• Recognize quantities having the same dimensions.
• Understand the uses and limitations of dimensional formulas.

Definition

For example, velocity is length divided by time: v = dt Therefore, its dimensional formula is: [v] = LT⁻¹

Introduction

Every physical quantity has a particular physical nature. For example: Length is associated with distance. Mass is associated with matter and inertia. Time is associated with duration. Velocity is associated with length divided by time. Force is associated with mass multiplied by acceleration. Dimensions provide a compact mathematical description of these relationships. The BIPM organizes physical quantities using seven independent SI base dimensions. Derived dimensions are formed by taking products of powers of these base dimensions. BIPM SI Brochure

Key Terms

Key Terms

Detailed Concept Explanation

5.1 Seven SI base dimensions

Seven SI base dimensions

The symbols L, M, and T represent dimensions, while m, kg, and s represent units.

5.2 General dimensional formula

The dimension of a general physical quantity Q can be written as: [Q] = MᵃLᵇTᶜIᵈΘᵉNᶠJᵍ Where: a = exponent of mass b = exponent of length c = exponent of time d = exponent of electric current e = exponent of temperature f = exponent of amount of substance g = exponent of luminous intensity The exponents may be: Positive Negative Zero Occasionally fractional NIST calls these powers dimensional exponents. NIST Guide to SI: Dimension of a Quantity

5.3 Meaning of a zero exponent

Consider velocity: [v] = M⁰LT⁻¹ Velocity does not depend on mass, so the exponent of M is zero. Since: M⁰ = 1 We normally write: [v] = LT⁻¹

5.4 Dimensions of derived quantities

Derived quantities obtain their dimensions from their defining equations. For force: F = ma Mass has dimension: [m] = M Acceleration has dimension: [a] = LT⁻² Therefore: [F] = M(LT⁻²) [F] = MLT⁻²

5.5 Dimensionless quantities

A quantity is dimensionless when all its dimensional exponents are zero: [Q] = M⁰L⁰T⁰ Therefore: [Q] = 1 Examples include: Strain Refractive index Relative density Coefficient of friction Plane angle Solid angle Efficiency A dimensionless quantity may still have a named unit. For example, plane angle may be expressed in radians.

Concept in Easy Words

A unit tells us the standard used for measurement. A dimension tells us the physical type of the measurement. For example, the distance between two places may be written in: metres kilometres centimetres The units are different, but the dimension is always: L Similarly, speed may be measured in metres per second or kilometres per hour, but its dimension remains: LT⁻¹ Changing the unit does not change the dimension.

Important Facts

Important Facts

• Every SI base quantity has its own base dimension.
• Dimension symbols are written using capital letters.
• Units and dimensions are not the same.
• Dimensions do not depend on the unit system.
• A dimensional formula is obtained from a physical equation.
• Numerical constants have no dimensions.
• Pure numbers such as 2 and π are dimensionless.
• Quantities being added or subtracted must have the same dimensions.
• Different physical quantities can have identical dimensions.
• A dimensionless quantity has all dimensional exponents equal to zero.
• Dimensions do not indicate whether a quantity is scalar or vector.
• Dimensions cannot determine numerical constants in an equation.

Formulas and Mathematical Explanation

8.1 Area A = lw [A] = [l][w] [A] = (L)(L) [A] = L²

8.2 Volume V = lwh [V] = [l][w][h] [V] = (L)(L)(L) [V] = L³

8.3 Velocity v = dt [v] = [d] / [t] [v] = LT [v] = LT⁻¹

8.4 Acceleration a = ΔvΔt [a] = [v] / [t] [a] = LT⁻¹ / T [a] = LT⁻²

8.5 Momentum p = mv [p] = [m][v] [p] = (M)(LT⁻¹) [p] = MLT⁻¹

8.6 Force F = ma [F] = [m][a] [F] = (M)(LT⁻²) [F] = MLT⁻²

8.7 Pressure P = FA [P] = [F] / [A] [P] = MLT⁻² / L² [P] = ML⁻¹T⁻²

8.8 Work and energy W = Fd [W] = [F][d] [W] = (MLT⁻²)(L) [W] = ML²T⁻²

8.9 Power P = Wt [P] = [W] / [t] [P] = ML²T⁻²T [P] = ML²T⁻³

8.10 Density ρ = m / V [ρ] = [m] / [V] [ρ] = M / L³ [ρ] = ML⁻³

8.11 Impulse J = FΔt [J] = [F][t] [J] = (MLT⁻²)(T) [J] = MLT⁻¹ Impulse and momentum have the same dimensions.

8.12 Electric charge Q = It [Q] = [I][t] [Q] = (I)(T) [Q] = IT

8.13 Gravitational constant Newton’s law of gravitation is: F = Gm₁m₂ / r² Rearrange for G: G = Fr² / (m₁m₂) Write dimensions: [G] = [F][r]² / ([m₁][m₂]) Substitute: [G] = (MLT⁻²)L²M² Simplify: [G] = ML³T⁻²M² [G] = M⁻¹L³T⁻²

Units, Symbols, and Dimensional Formulas

Quantity, SI unit, and dimensional formula

Unit-to-dimension replacement

Solved Examples

Real-Life Examples and Applications

Applications

• Dimensions help scientists check whether an equation is physically possible.
• Engineers use dimensional relationships when designing machines and structures.
• Unit-conversion systems depend on identifying the correct physical dimensions.
• Dimensions help compare formulas written in different unit systems.
• Computer programs use dimensional checking to detect unit errors.
• Dimensions help derive the possible form of relationships between physical quantities.
• Laboratory calculations use dimensions to verify final units.
• Dimensions help identify whether a result represents energy, force, pressure, or another quantity.

Comparison Tables

Dimensions versus units

Quantities with identical dimensions

Equal dimensions do not necessarily mean that the quantities are physically identical.

Key Logic

Use the following procedure: Write the defining physical equation. Ignore numerical constants such as 2, 12, and π. Replace each physical quantity with its dimensional formula. Apply the laws of indices. Combine the powers of M, L, T, and other dimensions. Write the final expression in standard dimensional form. Example: F = ma [F] = (M)(LT⁻²) [F] = MLT⁻²

Tips and Tricks

Remember the most important mechanical dimensions: Velocity: LT⁻¹ Acceleration: LT⁻² Momentum: MLT⁻¹ Force: MLT⁻² Energy: ML²T⁻² Power: ML²T⁻³ Pressure: ML⁻¹T⁻² Density: ML⁻³ A useful chain is: velocity = length / time acceleration = velocity / time force = mass × acceleration energy = force × distance power = energy / time

Common Mistakes

Common Mistakes

Topic Summary

The dimensions of a physical quantity show its dependence on base quantities. The seven SI base dimensions are represented by M, L, T, I, Θ, N, and J. Derived dimensional formulas are obtained from physical equations by replacing quantities with their base dimensions and simplifying their powers. Dimensions remain unchanged when units change. They help identify quantities and check physical equations, but they cannot distinguish every quantity or determine numerical constants.