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Dimensions of Physical Quantities
The dimension of a physical quantity shows how that quantity depends on the selected base quantities.
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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.
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.
Recommended for Class 11
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
| Term | Meaning |
|---|---|
| Dimension | Dependence of a quantity on base quantities |
| Base dimension | Dimension assigned to an SI base quantity |
| Derived dimension | Dimension formed from base dimensions |
| Dimensional formula | Expression showing powers of base dimensions |
| Dimensional exponent | Power of a base dimension |
| Dimensionless quantity | Quantity whose dimensional exponents are all zero |
| Unit | Agreed standard used to measure a quantity |
| Constant | Quantity whose value does not change under stated conditions |
| Variable | Quantity whose value can change |
| Homogeneity | Requirement that terms being added or equated have the same dimensions |
Detailed Concept Explanation
5.1 Seven SI base dimensions
Seven SI base dimensions
| Base quantity | Common quantity symbol | Dimension symbol | SI unit |
|---|---|---|---|
| Length | l, x, r | L | m |
| Mass | m | M | kg |
| Time | t | T | s |
| Electric current | I | I | A |
| Thermodynamic temperature | T | Θ | K |
| Amount of substance | n | N | mol |
| Luminous intensity | Iᵥ | J | cd |
The symbols L, M, and T represent dimensions, while m, kg, and s represent units.
5.2 General dimensional formula
- 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 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
| Quantity | SI unit | Dimensional formula |
|---|---|---|
| Length | m | L |
| Area | m² | L² |
| Volume | m³ | L³ |
| Mass | kg | M |
| Time | s | T |
| Velocity | m s⁻¹ | LT⁻¹ |
| Acceleration | m s⁻² | LT⁻² |
| Momentum | kg m s⁻¹ | MLT⁻¹ |
| Force | N | MLT⁻² |
| Pressure | Pa | ML⁻¹T⁻² |
| Work or energy | J | ML²T⁻² |
| Power | W | ML²T⁻³ |
| Density | kg m⁻³ | ML⁻³ |
| Frequency | Hz | T⁻¹ |
| Electric charge | C | IT |
| Plane angle | rad | 1 |
| Solid angle | sr | 1 |
Unit-to-dimension replacement
| SI unit | Corresponding dimension |
|---|---|
| kg | M |
| m | L |
| s | T |
| A | I |
| K | Θ |
| mol | N |
| cd | J |
Solved Examples
Example 1: Dimensions of kinetic energy
Find the dimensional formula of kinetic energy.
- Eₖ = (12)mv²
- The numerical constant 12 is dimensionless.
[Eₖ] = [m][v]²
[Eₖ] = (M)(LT⁻¹)²
[Eₖ] = (M)(L²T⁻²)
[Eₖ] = ML²T⁻²
Key step: A numerical constant does not affect dimensions.
Final answer: [Eₖ] = ML²T⁻²
Example 2: Dimensions of pressure
Derive the dimensional formula of pressure.
- P = FA
[P] = [F] / [A]
[P] = MLT⁻² / L²
[P] = ML¹⁻²T⁻²
[P] = ML⁻¹T⁻²
Final answer: [P] = ML⁻¹T⁻²
Example 3: Dimensions of power
Find the dimensional formula of power.
- P = Wt
[P] = [W] / [t]
[P] = ML²T⁻²T
[P] = ML²T⁻²⁻¹
[P] = ML²T⁻³
Final answer: [P] = ML²T⁻³
Example 4: Dimensions of spring constant
Using Hooke’s law, find the dimensions of the spring constant k.
- F = kx
- k = Fx
[k] = [F] / [x]
[k] = MLT⁻² / L
[k] = ML¹⁻¹T⁻²
[k] = MT⁻²
Final answer: [k] = MT⁻²
Example 5: Dimensions of coefficient of friction
Show that the coefficient of friction is dimensionless.
- μ = FfracfN
- [Ff] = MLT⁻²
- [N] = MLT⁻²
[μ] = MLT⁻² / MLT⁻²
[μ] = 1
The coefficient of friction is dimensionless.
Final answer: [μ] = M⁰L⁰T⁰
Example 6: Dimensions of surface tension
Surface tension is force per unit length. Find its dimensional formula.
- S = Fl
[S] = [F] / [l]
[S] = MLT⁻² / L
[S] = MT⁻²
Final answer: [S] = MT⁻²
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
| Feature | Dimension | Unit |
|---|---|---|
| Meaning | Physical nature of a quantity | Standard used to measure it |
| Example for length | L | m, cm or km |
| Example for time | T | s, min or h |
| Changes with unit system | No | Yes |
| Written using | Capital dimension symbols | Standard unit symbols |
| Example for velocity | LT⁻¹ | m s⁻¹ |
Quantities with identical dimensions
| Quantity 1 | Quantity 2 | Common dimensions |
|---|---|---|
| Work | Torque | ML²T⁻² |
| Momentum | Impulse | MLT⁻¹ |
| Pressure | Stress | ML⁻¹T⁻² |
| Frequency | Radioactive activity | T⁻¹ |
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
| Common mistake | Correct idea |
|---|---|
| Writing metre as the dimension of length | Metre is a unit; L is the dimension |
| Writing M for metre | M represents mass dimension |
| Including numerical constants in dimensions | Numerical constants are dimensionless |
| Writing velocity as LT | It is LT⁻¹ |
| Writing acceleration as LT⁻¹ | It is LT⁻² |
| Forgetting to square velocity in kinetic energy | Use [v]² = L²T⁻² |
| Assuming same dimensions mean same quantity | Different quantities can share dimensions |
| Using dimensions to determine numerical constants | Dimensional analysis cannot determine them |
| Adding quantities with different dimensions | Only like-dimensional quantities can be added |
| Confusing temperature symbol T with time dimension T | Temperature dimension is Θ |
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.
Exam Facts
Exam Facts
- A dimensional formula expresses a quantity using powers of base dimensions.
- The base dimensions include M, L, T, I, Θ, N, and J.
- Dimensional exponents may be positive, negative, or zero.
- A dimensionless quantity has all exponents equal to zero.
- Velocity has dimensions LT⁻¹.
- Acceleration has dimensions LT⁻².
- Momentum and impulse have dimensions MLT⁻¹.
- Force has dimensions MLT⁻².
- Work and energy have dimensions ML²T⁻².
- Power has dimensions ML²T⁻³.
- Pressure and stress have dimensions ML⁻¹T⁻².
- Density has dimensions ML⁻³.
- Dimensions remain unchanged when the measurement unit changes.
- Dimensional formulas cannot distinguish between all different physical quantities.
Exam Focus
Exam Focus
Important points
- A dimensional formula expresses a quantity using powers of base dimensions.
- The base dimensions include M, L, T, I, Θ, N, and J.
- Dimensional exponents may be positive, negative, or zero.
- A dimensionless quantity has all exponents equal to zero.
- Velocity has dimensions LT⁻¹.
- Acceleration has dimensions LT⁻².
- Momentum and impulse have dimensions MLT⁻¹.
- Force has dimensions MLT⁻².
- Work and energy have dimensions ML²T⁻².
- Power has dimensions ML²T⁻³.
- Pressure and stress have dimensions ML⁻¹T⁻².
- Density has dimensions ML⁻³.
- Dimensions remain unchanged when the measurement unit changes.
- Dimensional formulas cannot distinguish between all different physical quantities.
Important formulas
- [Q] = MᵃLᵇTᶜIᵈΘᵉNᶠJᵍ
- [v] = LT⁻¹
- [a] = LT⁻²
- [p] = MLT⁻¹
- [F] = MLT⁻²
- [P] = ML⁻¹T⁻²
- [W] = ML²T⁻²
- [Power] = ML²T⁻³
- [ρ] = ML⁻³
- [G] = M⁻¹L³T⁻²
- [k] = MT⁻²
- [S] = MT⁻²
Common exam traps
- Writing metre as the dimension of length
- Writing M for metre
- Including numerical constants in dimensions
- Writing velocity as LT
- Writing acceleration as LT⁻¹
- Forgetting to square velocity in kinetic energy
- Assuming same dimensions mean same quantity
- Using dimensions to determine numerical constants
- Adding quantities with different dimensions
- Confusing temperature symbol T with time dimension T
Long question structure
Dimensions describe the physical nature of a quantity. Each SI base quantity is assigned an independent base dimension. Mass, length, and time have dimensions M, L, and T. Derived dimensions are formed using products and powers of base dimensions. The dimensional formula of velocity is LT⁻¹. The dimensional formula of acceleration is LT⁻². Force is defined as mass multiplied by acceleration. Therefore, its dimensions are M multiplied by LT⁻². The dimensional formula of force is MLT⁻². Work is defined as force multiplied by displacement. Therefore, its dimensions are MLT⁻² multiplied by L. The dimensional formula of work is ML²T⁻². Power is defined as work divided by time. Therefore, its dimensions are ML²T⁻² divided by T. The dimensional formula of power is ML²T⁻³. Numerical constants do not appear in dimensional formulas. Dimensions remain the same in different systems of units. They help check equations and identify the nature of calculated quantities.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Dimensions show the physical nature of quantities.
Unit and dimension are different concepts.
Mass → M
Length → L
The dimension of a physical quantity shows how that quantity depends on the selected base quantities.
[Q] = MᵃLᵇTᶜIᵈΘᵉNᶠJᵍ; [v]=LT⁻¹; [a]=LT⁻²; [F]=MLT⁻²; [E]=ML²T⁻²; [P]=ML²T⁻³
- Definition
- The dimension of a physical quantity shows how that quantity depends on the selected base quantities.
- Formula
- [Q] = MᵃLᵇTᶜIᵈΘᵉNᶠJᵍ; [v]=LT⁻¹; [a]=LT⁻²; [F]=MLT⁻²; [E]=ML²T⁻²; [P]=ML²T⁻³
- Example
- Eₖ = (12)mv² → [Eₖ] = ML²T⁻²; μ = FfracfN → [μ] = 1
- Critical warning
- Writing metre as the dimension of length — Metre is a unit; L is the dimension
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Practice
Conceptual
Q1. Define the dimension of a physical quantity.
Conceptual
Q2. Differentiate between unit and dimension.
Conceptual
Q3. What is a dimensional exponent?
Conceptual
Q4. Define a dimensionless quantity and give three examples.
Conceptual
Q5. Can different quantities have the same dimensions? Give an example.
Conceptual
Q6. Write the seven SI base dimensions.
Conceptual
Q7. Why are numerical constants ignored in dimensional formulas?
Conceptual
Q8. State two limitations of dimensional formulas.
Derivation
Q9. Find the dimensions of area.
Derivation
Q10. Find the dimensions of volume.
Derivation
Q11. Find the dimensions of velocity.
Derivation
Q12. Find the dimensions of acceleration.
Derivation
Q13. Find the dimensions of momentum.
Derivation
Q14. Find the dimensions of force.
Derivation
Q15. Find the dimensions of pressure.
Derivation
Q16. Find the dimensions of energy.
Derivation
Q17. Find the dimensions of power.
Derivation
Q18. Find the dimensions of density.
Exam Practice
Important Short Question
Q1. What are dimensions of a physical quantity?
Long Question Preparation
Q2. Explain dimensions and derive the dimensional formulas of force, work, and power.