Deriving relations and changing units with dimensions

Dimensions can write a formula

You may not remember a formula, but you often know what it depends on. Dimensions can then find how it depends on them, apart from a pure number.

The method

The time $t$ for a ball to fall from rest through a height $h$ might depend on its mass $m$, the height $h$ and $g$.

1. Guess a product of powers. $t = k\,m^{x}h^{y}g^{z}$, where $k$ is a pure number.
2. Write the dimensions. $[\text{T}] = [\text{M}]^{x}[\text{L}]^{y}[\text{L}\,\text{T}^{-2}]^{z} = [\text{M}^{x}\,\text{L}^{y+z}\,\text{T}^{-2z}]$.
3. Match each power.
   • $\text{M}$: $0 = x$.
   • $\text{T}$: $1 = -2z$, so $z = -\frac{1}{2}$.
   • $\text{L}$: $0 = y + z$, so $y = \frac{1}{2}$.
4. Answer. $t = k\sqrt{\dfrac{h}{g}}$. The mass drops out! The true formula is $t = \sqrt{2h/g}$, so $k = \sqrt{2}$, which dimensions cannot find.

The same method gives a pendulum's period, $T = k\sqrt{l/g}$, with no mass in it. Check that with the bob.

You should find that the mass changes nothing, and that four times the length doubles the period, just as $\sqrt{l}$ says.

Changing units with dimensions

A quantity with dimensions $[\text{M}^a\,\text{L}^b\,\text{T}^c]$ has the number $n_1$ in one system and $n_2$ in another. The quantity itself does not change, so

Example. Change $1\ \text{J}$ ($[\text{M}\,\text{L}^2\,\text{T}^{-2}]$) to the CGS unit, the erg ($\text{g cm}^2\,\text{s}^{-2}$):

So $1\ \text{J} = 10^{7}\ \text{erg}$.

What dimensions cannot do

• They cannot find a pure number: the $2\pi$ or the $\sqrt{2}$.
• They cannot give a formula that is a sum of terms, like $s = ut + \frac{1}{2}at^2$.
• They cannot handle a formula with more unknown powers than base dimensions. In mechanics we have three equations (M, L, T), so at most three quantities.
• They cannot find a relation with a function like $\sin$ or $e^{x}$ in it.
• They cannot tell apart quantities with the same dimensions, like work and torque.

A worked example

The speed $v$ of waves on deep water might depend on the wavelength $\lambda$, the density of water $\rho$ and $g$. Find the form.

1. $v = k\,\lambda^{x}\rho^{y}g^{z}$: $[\text{L}\,\text{T}^{-1}] = [\text{L}]^{x}[\text{M}\,\text{L}^{-3}]^{y}[\text{L}\,\text{T}^{-2}]^{z}$.
2. M: $0 = y$. T: $-1 = -2z$, so $z = \frac{1}{2}$. L: $1 = x - 3y + z$, so $x = \frac{1}{2}$.
3. $v = k\sqrt{g\lambda}$. Longer waves run faster, and the density does not matter. (The true $k$ is $\dfrac{1}{\sqrt{2\pi}}$.)

Practise with new numbers

Try it in Python

The method is a set of equations, one for each of M, L and T. The program solves them with NumPy for the pendulum.

Remember these

• Write the result as $k\,a^{x}b^{y}c^{z}$ and match the powers of M, L and T.
• The pure number $k$ must come from theory or experiment.
• Change units with $n_2 = n_1 (M_1/M_2)^a (L_1/L_2)^b (T_1/T_2)^c$.
• No sums, no functions, no more than three unknown powers in mechanics.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The speed $v$ a body reaches after falling a height $h$ depends on $h$ and $g$. Find the form.

2. A pressure is $[\text{M}\,\text{L}^{-1}\,\text{T}^{-2}]$. How many dyne per square centimetre ($\text{g cm}^{-1}\,\text{s}^{-2}$) make one pascal?

3. Why can dimensions not derive $v^2 = u^2 + 2as$?

4. A pendulum on the Moon, where $g$ is about one sixth of the Earth's, swings with what period compared with the same pendulum on Earth?