Lessons · Engineering · checking an equation by its units
Let the units check the equation for you
Replace every symbol in an equation with its units and cancel. If what is left is not the unit the answer needs, the equation or a conversion is wrong.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
In a design review somebody asks where the 1000 came from, and the honest answer is that a kilonewton was left in a formula that wanted newtons. The units would have caught it in ten seconds, before the drawing went out.
How to think about it
Do the check with symbols and units only, no numbers. Cancel what appears on top and bottom. Compare the survivor with the unit you need, and if they differ, hunt for the missing conversion or the wrong power.
Worked example
F = m × aNewton's second law, symbols only.
[kg] × [m/s²] = kg·m/s² = NA newton is defined as exactly that, so the equation is at least dimensionally right.
P = F × v: N × m/s = N·m/s = J/s = WPower as force times speed. Joules per second is a watt, so this hangs together.
v = sqrt(2 g h): sqrt(m/s² × m) = sqrt(m²/s²) = m/s OKA square root of units is fine when what is inside is a perfect square. Here it is.
Your turn
Check the units of pressure, p = F / A, and name the unit that is left.
N / m² = (kg·m/s²) / m² = kg/(m·s²) =
Solve one, graded on the server
The trap
Believing a check that passes means the equation is right. Units cannot see a missing ½ or a 2 that should be a 4; they catch the wrong quantity, not the wrong constant. Passing the check is necessary, never sufficient.