Physics · Units and Measurements · Chapter notes
Unit Conversion and Dimensional Analysis · Class 11 Notes
Class 11 notes on unit conversion: SI base units and prefixes, dimensional analysis as a conversion engine, the dimensional formulae to know, and standard conversion tables.
In short
Converting a unit is multiplying by one. Write the conversion as a fraction equal to unity, arranged so the unit you want to lose cancels, and no power of ten can slip. Dimensional analysis is the same idea applied to a whole formula, which also lets you check an equation before you trust it.
Contents
Why Unit Conversion Matters
Every physical quantity is a number times a unit, and JEE routinely mixes SI, CGS and practical units in a single problem. Converting cleanly, and knowing when a conversion hides a factor of a hundred thousand, is a quiet source of easy marks and avoided mistakes.
The magnitude of a physical quantity is the same whatever unit you use, so n₁u₁ = n₂u₂: if the unit gets larger, the number gets smaller in proportion. To convert, multiply by the ratio of the old unit to the new one, written so the unwanted units cancel.
numeric value × unit = constant
SI Base Units and Prefixes
The whole SI system is built on seven base units; everything else is derived. The prefixes scale them by powers of ten.
| Quantity | SI unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
| Prefix | Factor | Prefix | Factor |
|---|---|---|---|
| giga (G) | 109 | milli (m) | 10−3 |
| mega (M) | 106 | micro (μ) | 10−6 |
| kilo (k) | 103 | nano (n) | 10−9 |
| centi (c) | 10−2 | pico (p) | 10−12 |
Dimensional Analysis: The Conversion Engine
The safest way to convert a derived unit is through its dimensions. Write the quantity in terms of M, L and T, then substitute the size of each base unit in the two systems.
1. Find the dimensional formula of the quantity (for force, [M L T−2], so a = 1, b = 1, c = −2).
2. Put in the ratio of each base unit, old over new.
3. Multiply out. The result is how many new units make one old unit.
Force has dimensions [M L T−2]. Going from SI (kg, m, s) to CGS (g, cm, s): 1 N = 1 × (1000 g / 1 g)(100 cm / 1 cm)(1)−2 = 1000 × 100 = 105 dyne. This factor of a hundred thousand is a classic trap.
Energy is [M L² T−2]. 1 J = (1000)(100)²(1) = 1000 × 10000 = 107 erg. Squaring the length ratio is what makes it ten million.
Dimensional Formulae You Must Know
Half of conversion and dimensional questions reduce to recalling these. Learn them until they are automatic.
| Quantity | Dimensional formula | SI unit |
|---|---|---|
| Velocity | [M0 L T−1] | m/s |
| Acceleration | [M0 L T−2] | m/s² |
| Force | [M L T−2] | newton (N) |
| Energy / Work | [M L² T−2] | joule (J) |
| Power | [M L² T−3] | watt (W) |
| Pressure | [M L−1 T−2] | pascal (Pa) |
| Momentum / Impulse | [M L T−1] | kg m/s |
| Density | [M L−3] | kg/m³ |
Standard Conversions Table
A reference set of the conversions JEE reuses most. Keep this handy.
| Quantity | Conversion |
|---|---|
| Force | 1 N = 105 dyne |
| Energy | 1 J = 107 erg |
| Energy (atomic) | 1 eV = 1.6 × 10−19 J |
| Energy (electrical) | 1 kWh = 3.6 × 106 J |
| Power | 1 HP = 746 W |
| Pressure | 1 atm = 1.013 × 105 Pa = 76 cm Hg |
| Pressure | 1 bar = 105 Pa |
| Length | 1 light year = 9.46 × 1015 m |
| Length | 1 Ångstrom = 10−10 m |
Most conversion mistakes come from a squared or cubed length. Area brings a factor of (100)² between m² and cm², and volume brings (100)³ between m³ and cm³. Always carry the exponent from the dimensional formula through the conversion.
Checking Equations by Dimensions
A bonus use of the same skill: the two sides of any correct physical equation must have identical dimensions. This catches wrong formulae instantly.
Every term added or equated in a physical equation must have the same dimensions. You cannot add a velocity to an acceleration. Use this to check a remembered formula, or to find the dimensions of an unknown constant.
Left side s is [L]. First term ut = (L T−1)(T) = [L]. Second term at² = (L T−2)(T²) = [L]. All three terms are [L], so the equation is dimensionally consistent. (Note: dimensional analysis cannot find pure numbers like the ½.)
Unit Conversion: One-Page Formula Sheet
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