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"Convert the gravitational constant into CGS units." "In a system where the unit of mass is α kg..." These questions look like they need a table of conversion factors. They need only the dimensional formula and one equation. This post sets out that equation, works through four examples including one where the base units themselves change, and lists the slips that make answers come out upside down.
Is this in the syllabus?
The JEE (Main) 2026 syllabus and the NEET (UG) 2026 syllabus both open physics with units of measurement, systems of units, dimensions of physical quantities and "dimensional analysis and its applications". The JEE (Advanced) 2026 brochure lists units and dimensions too. In the NCERT Class 11 chapter Units and Measurement (2026-27 reprint), section 1.6 names two applications, checking equations and deducing relations. Conversion appears in the exercises: Exercise 1.2 asks for conversions such as G into (cm)³ s⁻² g⁻¹, and Exercise 1.3 asks you to express a calorie in a system with mass unit α kg, length unit β m and time unit γ s. Check your year's syllabus.
What is the method?
A quantity is a number times a unit: Q = n₁u₁ = n₂u₂. The quantity is the same; only the description changes. If Q has dimensions [MᵃLᵇTᶜ], each unit is built from base units the same way, so
n₂ = n₁ × (M₁/M₂)ᵃ × (L₁/L₂)ᵇ × (T₁/T₂)ᶜ
where M₁, L₁, T₁ are the old base units and M₂, L₂, T₂ the new ones, written in the same measure. A sense check: a smaller unit gives a bigger number.
Worked example 1: pressure into CGS
The standard atmosphere is defined as 101 325 Pa, as NIST's table of conversion factors lists. Convert to dyn/cm².
Pressure has dimensions [ML⁻¹T⁻²], so a = 1, b = −1, c = −2.
M₁/M₂ = 1 kg / 1 g = 1000. L₁/L₂ = 1 m / 1 cm = 100. T₁/T₂ = 1.
n₂ = 101 325 × 1000¹ × 100⁻¹ = 1.013 25 × 10⁶.
So 1 atm = 1.013 × 10⁶ dyn/cm². The same NIST table gives 1 dyn/cm² = 0.1 Pa, which agrees.
Worked example 2: viscosity from CGS to SI
The poise is the CGS unit of viscosity, g cm⁻¹ s⁻¹. Viscosity has dimensions [ML⁻¹T⁻¹].
n₂ = 1 × (1 g / 1 kg)¹ × (1 cm / 1 m)⁻¹ = 10⁻³ × 10² = 0.1
So 1 poise = 0.1 Pa s, the value NIST lists.
Worked example 3: a made-up unit system
In a new system (sample units) the unit of mass is 2 kg, of length 0.5 m and of time 0.1 s. What is 1 J?
Energy has dimensions [ML²T⁻²].
n₂ = 1 × (1/2)¹ × (1/0.5)² × (1/0.1)⁻² = 0.5 × 4 × 0.01 = 0.02
Check by building the new unit directly: 2 kg × (0.5 m)² / (0.1 s)² = 50 J. One joule is 1/50 of that, which is 0.02. The method and the direct build agree.
Worked example 4: when the base quantities change
Some questions choose force, velocity and time as the base quantities. Say (sample units) the unit of force is 10 N, of velocity 5 m/s and of time 2 s. What are the units of mass, length and energy?
Write mass in terms of the new bases: [M] = [F]ˣ[v]ʸ[T]ᶻ. Matching powers of M, L and T gives x = 1, y = −1, z = 1, so mass = F × T / v.
Unit of mass = 10 × 2 / 5 = 4 kg.
Length = v × T, so the unit of length = 5 × 2 = 10 m.
Energy = force × length, so the unit of energy = 10 × 10 = 100 J.
This is the form most exam questions use. Solving for the exponents is the same skill as deducing a relation, which our post on dimensional analysis checks also practises.
Which mistakes cost marks?
Flipping the ratio to new over old. The answer then shrinks when it should grow.
Forgetting the power. cm² to m² is a factor of 10⁻⁴, not 10⁻².
Using dimensions for temperature scales. Celsius to kelvin is an offset, not a ratio; dimensions cannot do it.
Converting a dimensionless number. Strain, refractive index and angles in radians do not change.
Ignoring dimensional constants. G has dimensions [M⁻¹L³T⁻²], so its number changes between systems; a pure number like π does not.
If the dimensional formulae themselves are the gap, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can rebuild them before returning to conversions. For why the SI base units are defined as they are, see our post on the SI base units redefinition.
A practice routine
Day 1: Write the dimensional formulae for ten quantities from memory: force, energy, power, pressure, momentum, impulse, viscosity, surface tension, G and Planck's constant.
Day 2: Do NCERT Exercises 1.2 and 1.3 with the n₁u₁ = n₂u₂ method.
Day 3: Three made-up-system questions, checking each by building the new unit directly.
Day 4: Two questions with changed base quantities, like example 4.
Tonight, convert a surface tension of 0.072 N/m into dyn/cm, and check that the number gets bigger.
