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Dimensions and Dimensional Analysis

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Topics

Estimated time: 17 minutes
  • Definition: Dimensions
  • Application 1
  • Application 2
  • Application 3
  • Real-Life Applications
  • Limitations of Dimensional Analysis 
Maharashtra State Board: Class 11

Definition: Dimensions

Dimensions are the powers to which the fundamental units (like length, mass, time) must be raised to express any physical quantity.

Maharashtra State Board: Class 11

Application 1

Checking Equation Correctness 

The Golden Rule: Both sides of any valid physics equation must have the same dimensions. 

Real-Life Example: Checking if v = u + at is correct 

Left Side (LHS): 

  • v (final velocity) = [LT⁻¹]

Right Side (RHS): 

  • u (initial velocity) = [LT⁻¹]
  • at = acceleration × time = [LT−2] × [T] = [LT⁻¹]

Since both terms on RHS have dimension [LT⁻¹], and LHS also has [LT⁻¹], the equation is dimensionally correct.

Maharashtra State Board: Class 11

Application 2

Deriving Relationships: 

Real-World Problem: How does the time period of a pendulum depend on its length and gravity? 

Step 1: Identify the variables 

  • Time period: T depends on
  • Length: l and Acceleration due to gravity: g 

Step 2: Assume a relationship 
T = k × la × gb (where k is a dimensionless constant) 

Step 3: Write dimensions 

  • [T1] = [La] × [LT⁻²]b 
  • [T1] = [La+b T−2b]

Step 4: Compare powers 

  • For L: 0 = a + b → a = −b
  • For T: 1 = −2b → b = −1/2
  • Therefore: a = 1/2 

Step 5: Final relationship 
T = k × l¹/² × g⁻¹/² = k\[\sqrt \frac {l}{g}\]

(Experimentally, k = 2π) 

Maharashtra State Board: Class 11

Application 3

Unit Conversion 

Medical Example: Converting drug dosage from mg/kg to μg/g 

Problem: Convert 5 mg/kg to μg/g 

Step 1: Identify conversion factors 

  • 1 mg = 1000 μg
  • 1 kg = 1000 g 

Step 2: Apply dimensional analysis 
5\[\frac {mg}{kg}\] × \[\frac {1000μg}{1mg}\] × \[\frac {1kg}{1000g}\] = 5\[\frac {μg}{g}\]

Maharashtra State Board: Class 11

Real-Life Applications

  • Engineering (Bridge Safety): Engineers use dimensional analysis to ensure bridge design formulas are correct. A mistake in dimensions could lead to structural failure, affecting thousands of lives.
  • Medical (Drug Dosage): Doctors calculate medication doses using dimensional analysis. For a 70 kg patient needing 10 mg/kg of medicine: 
    70 kg × 10\[\frac {mg}{kg}\] = 700 mg total dose needed.
  • Space Missions (Fuel Calculations): NASA uses dimensional analysis to calculate fuel requirements. If engine thrust has dimensions [MLT⁻²] and specific impulse has dimensions [LT⁻¹], engineers can derive fuel consumption rates. 
Maharashtra State Board: Class 11

Limitations of Dimensional Analysis

  1. Cannot find dimensionless constants (like π, 2, 1/2 in formulas)
  2. Cannot handle trigonometric functions (sin, cos, tan are dimensionless)
  3. Cannot derive equations with multiple terms of the same dimension
  4. Cannot determine if equation is physically meaningful - only dimensionally consistent 

Example of Limitation: 
Both s = ut + \[\frac {1}{2}\]at2 and s = ut + 17at2 are dimensionally correct, but only the first is physically correct.

Maharashtra State Board: Class 11

Table: Some Common Physical Quantities their, SI Units and Dimensions

S. No. Physical Quantity Formula SI Unit Dimensional Formula
1 Density ρ = M/V kilogram per cubic metre (kg/m³) [L⁻³ M¹ T⁰]
2 Acceleration a = v/t metre per second square (m/s²) [L¹ M⁰ T⁻²]
3 Momentum P = mv kilogram metre per second (kg·m/s) [L¹ M¹ T⁻¹]
4 Force F = ma kilogram metre per second square (kg·m/s²) or newton (N) [L¹ M¹ T⁻²]
5 Impulse J = F·t newton second (N·s) [L¹ M¹ T⁻¹]
6 Work W = F·s joule (J) [L² M¹ T⁻²]
7 Kinetic Energy KE = ½ mv² joule (J) [L² M¹ T⁻²]
8 Pressure P = F/A kilogram per metre second square (kg/m·s²) or pascal (Pa) [L⁻¹ M¹ T⁻²]

Test Yourself

Related QuestionsVIEW ALL [12]

A length-scale (l) depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression for l is dimensionally correct?.

[L1M1T-2] is the dimensional formula for ______.

If the force is proportional to the square of velocity, then the dimension of the proportionality constant is ______

The dimensional formula for gravitational constant G is ______

Write a short note on the following.

Dimensionless quantities

The velocity of a particle v at an instant t is given by v = at + bt2. The dimensions of b are ______ 

The dimension of `(mu_0ε_0)^{1/2}` is ______

Assuming that the frequency γ of a vibrating string may depend upon i) applied force (F) ii) length (l) iii) mass per unit length (m), prove that ϒ ∝ `1/lsqrt(F/m)` using dimensional analysis.

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