Physics › Maths bridge › Rearranging equations
Rearranging equations
Changing the subject of an equation, reliably, including through squares and roots.
One rule, applied in reverse order
Apply the same inverse operation to both sides. Work from the outermost operation towards the subject, checking each line by substitution or units. Adding undoes subtracting, dividing undoes multiplying, and a square root undoes a square.
A formula triangle holds only three symbols, so it cannot be applied to most A-level equations. The both-sides method works for an equation of any size; practise it on short rearrangements first.
Through brackets, squares and roots
WORKED EXAMPLE
A subject inside a bracket
Solve ε = I(R + r) for r.
The r sits inside a bracket that is multiplied by I. Undo the outermost layer first, so divide both sides by I, giving ε/I = R + r.
Now subtract R and you have r = ε/I − R. Two moves, each applied to both sides, outermost first.
WORKED EXAMPLE
When the subject is under a root
The pendulum equation is T = 2π√(l/g). Solve it for g.
The outermost operation on the right is the multiplication by 2π, so divide both sides by 2π first, giving T/(2π) = √(l/g).
Squaring both sides removes the root: T2/(4π2) = l/g.
Multiply both sides by g and by 4π2/T2, and g = 4π2l/T2.
A pendulum of length 0.99 m with a period of 2.0 s gives g = 4π2 × 0.99/2.02 = 9.77 m s−2, close to the accepted 9.81 m s−2, which supports the rearrangement.
GUIDED PRACTICE
A rearrangement and a units check
Solve v2 = u2 + 2as for a, then check that the units of the result make sense.
Show the working
Subtract u2 from both sides, then divide by 2s: a = (v2 − u2)/(2s).
Now check the units. (m s−1)2 divided by metres is m2 s−2/m = m s−2, an acceleration. Algebra and units agree, so the rearrangement stands.
Where physics leans on this: EMF and internal resistance · Motion graphs and SUVAT · Pendulums and springs. All eight skills: the maths bridge.