One table in the school laboratory gives the surface tension of water as \(0.072\,\text{N/m}\), a force per unit length. A reference book on the next shelf gives \(0.072\,\text{J/m}^2\), an energy per unit area. A force is not an energy, and a length is not an area. Is one of the two books wrong?
What a quantity is made of
Every derived quantity is built from the base quantities by its defining equation. Speed is a length divided by a time. Acceleration is a speed divided by a time. Force is a mass times an acceleration. The recipe of powers that builds a quantity from mass, length and time is called its dimensions. We write the dimensions of a quantity \(Q\) as \([Q]\), with M for mass, L for length and T for time.
Follow the recipe of force. Speed is \([v] = \text{L}/\text{T} = \text{LT}^{-1}\). Acceleration is a speed per time, \([a] = \text{LT}^{-2}\). Force is mass times acceleration, \([F] = \text{MLT}^{-2}\). Work is force times distance, \([W] = \text{ML}^2\text{T}^{-2}\). Power is work per time, \([P] = \text{ML}^2\text{T}^{-3}\). The expression on the right of each line is the dimensional formula of the quantity.
The dimensions of a physical quantity are the powers to which the base quantities must be raised to build it. In mechanics the base quantities are mass, length and time, so every dimensional formula has the form \(\text{M}^a\text{L}^b\text{T}^c\). Heat and electricity add temperature K and current A.
Some quantities have no dimensions at all. A ratio of two lengths, such as a strain, is a pure number. An angle in radians is an arc length divided by a radius, so it has no dimensions either. Refractive index and relative density are ratios too. Such quantities are dimensionless, and their dimensional formula is \(\text{M}^0\text{L}^0\text{T}^0\).
The slope and the area of a graph have dimensions too. A slope divides the vertical quantity by the horizontal one. An area multiplies them. On a velocity-time graph, Fig 4.1, the slope is \(\text{LT}^{-1}/\text{T} = \text{LT}^{-2}\), an acceleration, and the area is \(\text{LT}^{-1} \times \text{T} = \text{L}\), a displacement.
Pressure is force per unit area. What are the dimensions of pressure?
Which of these quantities are dimensionless?
Like with like
A length can be added to a length, but never to a time. Three metres plus two seconds has no meaning. So in any correct physical equation, every term that is added, subtracted or set equal to another has the same dimensions. This rule is the principle of homogeneity.
In a correct physical equation, every term has the same dimensions. The two sides of the equation, and every quantity added to or subtracted from another, must have one dimensional formula.
Why must nature obey this rule? A law of physics cannot depend on the units we choose. A car moves the same distance whether we measure it in metres or in centimetres. A homogeneous equation gives one physical answer in every system of units. An equation that mixes dimensions gives different answers in different units, so it cannot describe nature. Fig 4.2 tests a correct and a wrong equation this way.
The wrong equation fails because \(at^3\) has dimensions \(\text{LT}^{-2} \times \text{T}^3 = \text{LT}\), not L. When the unit of time changes, that term changes by a different factor from the others. A term with the right dimensions scales exactly like the left side, so the correct equation survives every change of unit.
A student writes the displacement of a body with uniform acceleration as \(s = ut + \tfrac12 at^3\). What does a check of dimensions show?
Checking an equation
To check an equation, write the dimensions of every term, drop the pure numbers, and compare. For \(v^2 = u^2 + 2as\), the terms give \(\text{L}^2\text{T}^{-2}\), \(\text{L}^2\text{T}^{-2}\) and \(\text{LT}^{-2} \times \text{L} = \text{L}^2\text{T}^{-2}\). All three match, so the equation passes. The same idea finds the dimensions of an unknown constant. In \(F = a + bt\), the term \(bt\) is added to a force, so \([b] = \text{MLT}^{-2}/\text{T} = \text{MLT}^{-3}\).
A function such as a sine, an exponential or a logarithm can act only on a pure number. So in a wave \(y = A\sin(\omega t - kx)\), both \(\omega t\) and \(kx\) must be dimensionless. Then \([\omega] = \text{T}^{-1}\) and \([k] = \text{L}^{-1}\), and the ratio \(\omega/k\) has dimensions \(\text{LT}^{-1}\), a speed. The amplitude \(A\) carries the length.
The speed of a wave on a string is \(v = \sqrt{T/\mu}\), where \(T\) is the tension and \(\mu\) the mass per unit length. First check that the formula gives a speed. Then find the speed on a guitar string \(64\,\text{cm}\) long, of mass \(2.0\,\text{g}\), under a tension of \(80\,\text{N}\).
Dimensions: \([T] = \text{MLT}^{-2}\) and \([\mu] = \text{M}/\text{L} = \text{ML}^{-1}\). So \([T/\mu] = \text{MLT}^{-2}/\text{ML}^{-1} = \text{L}^2\text{T}^{-2}\), and its square root is \(\text{LT}^{-1}\), a speed. The formula passes.
Units: \(L = 0.64\,\text{m}\) and \(m = 2.0\,\text{g} = 0.0020\,\text{kg}\), so \(\mu = 0.0020/0.64 = 0.003125\,\text{kg/m}\).
Speed: \(v = \sqrt{80/0.003125} = \sqrt{25\,600} = 160\,\text{m/s}\), the answer of the physics chapter.
The phenomenon is answered. A force per length has dimensions \(\text{MLT}^{-2}/\text{L} = \text{MT}^{-2}\). An energy per area has \(\text{ML}^2\text{T}^{-2}/\text{L}^2 = \text{MT}^{-2}\), the same. So N/m and J/m² are one unit, since \(1\,\text{J} = 1\,\text{N m}\). Both books are right, and the chapter on surface tension explains the physics behind the two readings.
Surface tension has dimensions \(\text{MT}^{-2}\). Which of these has the same dimensions?
A force changes with time as \(F = a + bt\). What are the dimensions of \(b\)?
A dimensional check can prove an equation wrong, but it can never prove one right. It cannot see a wrong pure number: \(s = ut + 2at^2\) passes as well as the true equation. It cannot see a missing term of the right dimensions. And two different quantities can share one dimensional formula, as work and torque do, so a match does not tell you which one you have.
Joseph Fourier wrote the first great book on the flow of heat, The Analytical Theory of Heat. In it he noticed that every term of his equations carried the same powers of length, time and temperature. He called these powers the dimensions of the quantities. He wrote that an equation whose terms had different dimensions must contain an error of calculation. Engineers and physicists have used his test ever since.
The chapter on surface tension shows that N/m and J/m² are one unit, with dimensions \(\text{MT}^{-2}\). The viscosity chapter finds the dimensions \(\text{ML}^{-1}\text{T}^{-1}\) of the viscosity. The waves chapter checks that \(\sqrt{T/\mu}\) is a speed. Bernoulli's equation adds a pressure to an energy per volume, and the RC circuit's time constant is shown to be a time. Calorimetry and heat conduction use the same check on their units.
- Believing an equation because its dimensions balance.The check can only reject. A wrong pure number or a missing term passes it.
- Giving dimensions to a pure number such as \(\tfrac12\) or \(2\pi\).Pure numbers are dimensionless. Drop them before comparing.
- Leaving a unit such as grams inside a formula written for SI units.Convert to kg, m and s first; \(2.0\,\text{g}\) is \(0.0020\,\text{kg}\).
- Writing \(\text{MLT}^{-2}\) for pressure.That is force. Pressure divides force by an area: \(\text{ML}^{-1}\text{T}^{-2}\).