Dimensional Formulae of Common Quantities (Full List)
Physics · Units And Measurements · NEET
A dimensional formula shows a quantity in terms of the seven base dimensions, mostly [M], [L] and [T] for mechanics. For example velocity is [M⁰ L¹ T⁻¹], force is [M¹ L¹ T⁻²], and energy is [M¹ L² T⁻²]. Memory hook: never memorise the whole list. Learn one root formula (like F = ma), then build every other quantity from its defining equation.
Do not memorise a table. Start from one defining equation (F = ma) and chain-multiply to reach velocity, force, work and power, building each dimensional formula step by step.
Your doubts, answered
How do I write the dimensional formula of any quantity without memorising a table?
Start from the defining equation and replace each quantity by its dimensions. Force = mass × acceleration, so [F] = [M] × [L T⁻²] = [M L T⁻²]. Work = force × distance, so [W] = [M L T⁻²] × [L] = [M L² T⁻²]. If you know a quantity's formula, you can always build its dimensions. This is why NEET rewards understanding over memory.
Do momentum and impulse have the same dimensional formula?
Yes. Momentum p = m·v = [M][L T⁻¹] = [M L T⁻¹]. Impulse J = Force × time = [M L T⁻²][T] = [M L T⁻¹]. Both are [M L T⁻¹]. Whenever two quantities are physically equal (impulse equals change in momentum), their dimensions must match. NEET often tests such pairs.
Which common quantity has the dimension M⁻¹ (reciprocal of mass)?
The gravitational constant G. From F = G·m₁m₂/r², G = F·r²/(m₁m₂) = [M L T⁻²][L²]/[M²] = [M⁻¹ L³ T⁻²]. It carries M to the power −1, so it is the standard 'reciprocal of mass' quantity NEET asks about. Torque and angular momentum both carry positive powers of mass, so they are wrong.
What is the difference between dimensional formula and unit?
A unit is a specific scale of measurement (newton, joule), while a dimensional formula shows the base-quantity make-up ([M L T⁻²], [M L² T⁻²]). Two quantities can share dimensions but not meaning, for example work and torque are both [M L² T⁻²] yet one is energy and the other a turning effect. Dimensions tell you structure, not identity.
Why do angle, strain and refractive index show no dimensions?
They are ratios of two same-kind quantities, so the dimensions cancel. Angle = arc/radius = [L]/[L], strain = change in length/length = [L]/[L], refractive index = speed/speed. Each is [M⁰ L⁰ T⁰], a dimensionless number. NEET likes to group these as 'quantities with unit but no dimension' (like angle) versus 'no unit and no dimension' (like strain, refractive index).
⚠️ The NEET trap ✗ Work and torque have different dimensional formulae because one is energy and the other is a moment. ✓ Both work and torque are [M L² T⁻²]. Torque = force × perpendicular distance = [M L T⁻²][L] = [M L² T⁻²], the same as work. Dimensions cannot separate them, only the physics (scalar energy vs vector moment) does. 🧠 Same dimensions never means same quantity. NEET plants pairs like work-torque and pressure-stress-energy-density to see if you blindly match by [M L² T⁻²].
Real NEET questions
2022
Match List-I with List-II. List-I: (a) Gravitational constant G (b) Gravitational potential energy (c) Gravitational potential (d) Gravitational intensity. List-II: (i) [L²T⁻²] (ii) [M⁻¹L³T⁻²] (iii) [LT⁻²] (iv) [ML²T⁻²]. Choose the correct match:
A · (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
B · (a)-(ii), (b)-(iv), (c)-(i), (d)-(iii) ✓
C · (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)
D · (a)-(iv), (b)-(ii), (c)-(i), (d)-(iii)
Solution: Build each from its defining equation. G from F = Gm₁m₂/r²: G = F·r²/(m₁m₂) = [M L T⁻²][L²]/[M²] = [M⁻¹ L³ T⁻²] → (ii). Gravitational potential energy is energy = [M L² T⁻²] → (iv). Gravitational potential = PE per unit mass = [M L² T⁻²]/[M] = [L² T⁻²] → (i). Gravitational intensity = force per unit mass = [M L T⁻²]/[M] = [L T⁻²] → (iii). So (a)-ii, (b)-iv, (c)-i, (d)-iii, which is option B.
2022
The dimensions [M L T⁻² A⁻²] belong to the:
A · Magnetic flux
B · Self inductance
C · Magnetic permeability ✓
D · Electric permittivity
Solution: Use a defining relation. For a long solenoid B = μ₀nI, and force per length between wires F/L = μ₀I₁I₂/(2πd). So μ₀ = (F/L)·d/(I₁I₂) = [M L T⁻²]·[L⁻¹]·[L]/[A²] = [M L T⁻² A⁻²]. That matches the given dimensions, so it is magnetic permeability μ₀. (Check the trap: self-inductance is [M L² T⁻² A⁻²], one extra L², so option B is wrong.)
2023
The mechanical quantity which has dimensions of the reciprocal of mass (M⁻¹) is:
A · Torque
B · Gravitational constant ✓
C · Angular momentum
D · Coefficient of thermal conductivity
Solution: Test each for the power of M. Torque = force × distance = [M L² T⁻²] (M¹). Angular momentum = mvr = [M L² T⁻¹] (M¹). Thermal conductivity = [M L T⁻³ K⁻¹] (M¹). Gravitational constant G = F·r²/(m₁m₂) = [M L T⁻²][L²]/[M²] = [M⁻¹ L³ T⁻²]. Only G carries M to the power −1, so the answer is the gravitational constant, option B.
Solved Units And Measurements NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
What is the dimensional formula of velocity, acceleration and force?
Velocity = displacement/time = [L]/[T] = [M⁰ L T⁻¹]. Acceleration = velocity/time = [L T⁻¹]/[T] = [M⁰ L T⁻²]. Force = mass × acceleration = [M][L T⁻²] = [M L T⁻²].
What are the dimensional formulae of energy, power and pressure?
Energy (work) = force × distance = [M L² T⁻²]. Power = energy/time = [M L² T⁻³]. Pressure = force/area = [M L T⁻²]/[L²] = [M L⁻¹ T⁻²], the same as stress.
Which quantities are dimensionless?
Angle, solid angle, strain, refractive index, relative density, and pure numbers like 2π. They are ratios of same-kind quantities, so all base powers are zero: [M⁰ L⁰ T⁰]. Note angle has a unit (radian) but no dimension.
Do pressure, stress, Young's modulus and energy density share dimensions?
Yes. All four are [M L⁻¹ T⁻²]. Pressure and stress are force/area; Young's modulus is stress/strain and strain is dimensionless; energy density is energy/volume = [M L² T⁻²]/[L³]. NEET frequently exploits this cluster.
How is the dimensional formula list useful for NEET?
It lets you check equations for consistency, convert units, and solve 'which quantity has these dimensions' problems that appear almost every year. But do not rote-memorise; learn to derive from defining equations so you can handle any new quantity in the exam.
What are the dimensions of Planck's constant h and gravitational constant G?
From E = hf, h = E/f = [M L² T⁻²]/[T⁻¹] = [M L² T⁻¹]. From F = Gm₁m₂/r², G = [M⁻¹ L³ T⁻²]. Both are high-value NEET quantities and appear in fundamental-constant combination problems.
