Physics 1st Paper · Classes 11-12 · Chapter 1 · 1.6

(part) Dimensions and Dimensional Equations.

You can often tell whether an equation is right without putting in a single number, just by comparing the dimensions on its two sides. That is the move exams ask for most. So first you learn the dimension of each quantity, then what you can do with them.

1. What a dimension is

The symbol we use to link a quantity to its fundamental units is the dimension of that quantity. Put another way: to build a derived unit you raise fundamental units to certain powers, and those powers are the dimensions of that derived unit.

The dimensions of length, mass and time are L, M and T. Area carries length twice, so it is L²; volume carries it three times, so it is L³. Square brackets [ ] mean “the dimension of”.

An equation that relates fundamental and derived units is a dimensional equation. For example, velocity = displacement ÷ time, so [v] = L/T = [LT⁻¹].

2. Dimensions of mechanical quantities

Velocity is [LT⁻¹], acceleration [LT⁻²], force [MLT⁻²]. Momentum and impulse both have [MLT⁻¹].

Pressure, stress and elastic modulus share one dimension, [ML⁻¹T⁻²]. Density is [ML⁻³], the coefficient of viscosity [ML⁻¹T⁻¹], surface tension [MT⁻²].

Work, energy, moment of force and torque all have [ML²T⁻²]. Power is [ML²T⁻³], which is work ÷ time.

The rotation quantities: moment of inertia [ML²], angular momentum [ML²T⁻¹], radius of gyration [L]. Frequency and angular velocity are both [T⁻¹], angular acceleration is [T⁻²].

Gravitational field strength and acceleration due to gravity share [LT⁻²]. Gravitational potential is [L²T⁻²] and the velocity gradient is [T⁻¹].

3. Dimensions in heat, electricity and magnetism

Temperature has dimension θ. Specific heat is [L²T⁻²θ⁻¹], latent heat [L²T⁻²]. Heat capacity, entropy and the Boltzmann constant share [ML²T⁻²θ⁻¹].

Thermal conductivity is [MLT⁻³θ⁻¹], the Stefan constant [MT⁻³θ⁻⁴], temperature gradient [θL⁻¹], the molar gas constant [ML²T⁻²θ⁻¹mol⁻¹].

In electricity the symbol for current is I. Electric charge is [IT], current density [IL⁻²], electric dipole moment [ITL].

Electric potential is [ML²T⁻³I⁻¹], electric field strength [MLT⁻³I⁻¹], resistance [ML²T⁻³I⁻²], resistivity [ML³T⁻³I⁻²].

Capacitance is [M⁻¹L⁻²T⁴I²] and inductance is [ML²T⁻²I⁻²].

Magnetic flux is [ML²T⁻²I⁻¹], magnetic flux density [MT⁻²I⁻¹], magnetic dipole moment [IL²], magnetic pole strength [IL].

Last two: the intensity of a wave is [MT⁻³], surface energy is [MT⁻²].

Before memorising, play it backwards. Set the powers of M, L and T yourself and see which quantities have that dimension.

Before you memorise, test yourself. Sixteen quantities in four families: pick a quantity, then tap its dimension. A wrong pair shows below how the right dimension comes out of the relation.

4. Quantities with no dimension

Some quantities have no dimension at all. These are dimensionless: angle, solid angle, strain, Poisson's ratio and relative density.

Why? Plane angle θ = arc ÷ radius = [L]/[L] = 1. Length cancels length and nothing is left, so the plane angle is dimensionless.

5. The homogeneity principle and what dimensions are for

The principle of dimensional homogeneity: in a correct relation or equation the dimensions of the two sides are always the same. If the two sides do not match, the equation is wrong, and you need no numbers to see it.

Dimensional equations are used in five ways: converting units from one system to another, checking an equation, forming an equation, finding the unit of a quantity, and solving physical problems.

The principle is a balance: each side's dimension goes on a pan, and if they differ the beam tips.

6. Deriving a dimension

The book's Example 1.2 has three parts. (1) E = F/q with [F] = MLT⁻² and [q] = IT, so [E] = MLT⁻²/IT = [MLT⁻³I⁻¹] (the book writes A for current). (2) Angular momentum mvr = [M][LT⁻¹][L] = [ML²T⁻¹], and Planck's constant h = E/ν = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹], so the two share a dimension. (3) With force (F), length (L) and time (T) as base quantities, mass = force ÷ acceleration = F/LT⁻² = [FL⁻¹T²]. Every time it is four steps: write the relation, put the dimensions in, simplify, state the answer. Open the steps yourself; pressure is there too.

7. Checking an equation

Now put the principle to work, in Example 1.4. (1) v = √(P/D) is dimensionally correct, because the right side √(ML⁻¹T⁻²/ML⁻³) = [LT⁻¹] and the left side, a velocity, is also [LT⁻¹]. (2) Fs = ½mv² − ½mu² is correct because both sides are [ML²T⁻²]. The method never changes: work out the two sides separately, then compare. Put the steps of the first one in order.

The second is the work and energy equation. Two terms are being subtracted, so their dimensions have to match.

8. Check yourself

Three questions in the exam's style. Start with the first.

The second makes you derive a dimension from a relation.

In the third, watch the signs of the powers.

The whole note on eight cards.

Source: Physics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Physical World and Measurement”, section 1.6.