Physics and Measurement

Complete Academic Reference of Units, Dimensions, and Measurement Principles

Units & Systems
Dimensions & Formulas
Same Dimensions
Applications
Thermodynamics
Analysis Tools

1. Units and Measurement

Unit: A standard used for the measurement of a physical quantity.

Any physical quantity can be expressed as: Q = n × u

Where n represents the numerical value and u represents the unit of measurement.

1.1 System of Units

The International System of Units (SI) was established by the General Conference on Weights and Measures in 1971 and has been internationally adopted. This system comprises seven base units along with two supplementary units.

1.2 Base Units (Fundamental Units)

These units form the foundation for all measurements in physics:

Length
meter (m)
Standard of length
Mass
kilogram (kg)
Standard of mass
Time
second (s)
Standard of time
Electric Current
ampere (A)
Standard of electric current
Temperature
kelvin (K)
Standard of temperature
Luminous Intensity
candela (cd)
Standard of luminous intensity
Amount of Substance
mole (mol)
Standard of amount of substance

1.3 Derived Units

Derived units are expressed as combinations of fundamental units. Examples include:

  • Velocity (m/s)
  • Force (Newton, N = kg·m/s²)
  • Impulse (N·s)
  • Work (Joule, J = N·m)
  • Power (Watt, W = J/s)

2. Dimensions and Dimensional Formulas

All physical quantities can be expressed in terms of the seven fundamental dimensions, denoted within square brackets [ ].

2.1 Dimensions of Physical Quantities (Mechanics)

S.No. Physical Quantity Definition/Formula SI Unit Dimensional Formula
1 Velocity Displacement / Time m/s [M⁰LT⁻¹]
2 Acceleration Velocity / Time m/s² [M⁰LT⁻²]
3 Force Mass × Acceleration N (kg·m/s²) [MLT⁻²]
4 Work Force × Displacement J (N·m) [ML²T⁻²]
5 Energy Capacity to do work J [ML²T⁻²]
6 Torque Force × Perpendicular distance N·m [ML²T⁻²]
7 Power Work / Time W (J/s) [ML²T⁻³]
8 Momentum Mass × Velocity kg·m/s [MLT⁻¹]
9 Impulse Force × Time N·s [MLT⁻¹]
10 Angle Arc / Radius radian (no unit) [M⁰L⁰T⁰]
11 Strain ΔL/L or ΔV/V (no unit) [M⁰L⁰T⁰]
12 Stress Force / Area N/m² [ML⁻¹T⁻²]
13 Pressure Force / Area N/m² [ML⁻¹T⁻²]
14 Modulus of Elasticity Stress / Strain N/m² [ML⁻¹T⁻²]
15 Frequency 1 / Time period Hz (s⁻¹) [M⁰L⁰T⁻¹]
16 Angular Velocity Angle / Time rad/s [M⁰L⁰T⁻¹]
17 Moment of Inertia Mass × (Distance)² kg·m² [ML²T⁰]
18 Surface Tension Force / Length N/m [ML⁰T⁻²]
19 Gravitational Constant F × r² / m² N·m²/kg² [M⁻¹L³T⁻²]

3. Physical Quantities with Identical Dimensions

[M⁰L⁰T⁻¹]

Frequency, angular frequency, angular velocity, velocity gradient, decay constant

[ML²T⁻²]

Work, internal energy, potential energy, kinetic energy, torque, moment of force

[ML⁻¹T⁻²]

Pressure, stress, Young's modulus, bulk modulus, modulus of rigidity, energy density

[MLT⁻¹]

Momentum, impulse

[M⁰LT⁻²]

Acceleration due to gravity, gravitational field intensity

[ML²T⁻¹]

Angular momentum, Planck's constant

[MT⁻²]

Surface tension, surface energy (energy per unit area)

[M⁰L⁰T⁰]

Strain, refractive index, relative density, angle, solid angle, relative permittivity, relative permeability

[M⁰L²T⁻²]

Latent heat, gravitational potential

[ML²T⁻²K⁻¹]

Thermal capacity, gas constant, Boltzmann constant, entropy

3.1 Special Dimension Groups

[M⁰L⁰T⁻¹]

√(l/g), √(m/k), √(R/g)

Where l = length, g = acceleration due to gravity, m = mass, k = spring constant, R = Radius of earth

[M⁰T⁻¹]

L/R, √(LC), RC

Where L = inductance, R = resistance, C = capacitance

[ML²T⁻²]

I²Rt, V²t/R, VIt, qV, LI², q²/C, CV²

Where I = current, t = time, q = charge, L = inductance, C = capacitance, R = resistance, V = voltage

4. Applications of Dimensional Analysis

Conversion of Units

Transforming measurements between different unit systems

Dimensional Verification

Checking the correctness of physical equations

Relationship Derivation

Establishing connections between physical quantities

Constant Dimensioning

Determining dimensions of physical constants

4.1 Least Count

The least count of a measuring instrument is the smallest value that can be measured using that instrument. This parameter is significant for two reasons:

  • It indicates the permissible error in measurement
  • It defines the resolution limit of the instrument

5. Thermodynamics Quantities and Dimensions

S.No. Physical Quantity SI Unit Dimensional Formula
1 Thermodynamic Temperature kelvin (K) [M⁰L⁰T⁰K]
2 Heat joule (J) [ML²T⁻²]
3 Specific Heat J·kg⁻¹·K⁻¹ [M⁰L²T⁻²K⁻¹]
4 Latent Heat J·kg⁻¹ [M⁰L²T⁻²]
5 Universal Gas Constant J·mol⁻¹·K⁻¹ [ML²T⁻²K⁻¹mol⁻¹]
6 Boltzmann's Constant J·K⁻¹ [ML²T⁻²K⁻¹]
7 Stefan's Constant J·s⁻¹·m⁻²·K⁻⁴ [MT⁻³K⁻⁴]
8 Planck's Constant J·s [ML²T⁻¹]
9 Solar Constant J·m⁻²·s⁻¹ [ML⁰T⁻³]
10 Thermal Conductivity J·s⁻¹·m⁻¹·K⁻¹ [MLT⁻³K⁻¹]
11 Thermal Resistance K·s·cal⁻¹ [M⁻¹L⁻²T³K]
12 Enthalpy cal [ML²T⁻²]
13 Entropy cal·K⁻¹ [ML²T⁻²K⁻¹]

6. Dimensional Analysis Tools

Dimensional Analysis Verification

Use this tool to verify the dimensional consistency of physical equations.

Note: This tool provides a basic dimensional analysis. For complex expressions, manual verification may be required.

6.1 Common Dimensional Formulas Reference

Mechanics

  • Velocity: [M⁰LT⁻¹]
  • Acceleration: [M⁰LT⁻²]
  • Force: [MLT⁻²]
  • Work/Energy: [ML²T⁻²]
  • Power: [ML²T⁻³]
  • Momentum: [MLT⁻¹]
  • Impulse: [MLT⁻¹]

Thermodynamics

  • Heat: [ML²T⁻²]
  • Specific Heat: [M⁰L²T⁻²K⁻¹]
  • Entropy: [ML²T⁻²K⁻¹]
  • Thermal Conductivity: [MLT⁻³K⁻¹]