Physics › Maths bridge › Proportional reasoning
Proportional reasoning
Direct, inverse and square relationships, and answering doubling questions without a calculator.
Reading a relationship
y ∝ x means doubling x doubles y. y ∝ x2 means doubling x quadruples y. y ∝ 1/x means doubling x halves y, and the inverse square, y ∝ 1/x2, means doubling x cuts y to a quarter. Each of these gives a one-line answer to a question that appears to require a full calculation.
One move covers all of them. Write the relationship, apply the scale factor to the right-hand side, and read off the effect on the left. No values are needed, so the question tests whether the structure of the relationship is understood rather than the arithmetic.
Scaling without a calculator
WORKED EXAMPLE
An inverse-square jump
The gravitational force between two masses follows F ∝ 1/r2. The separation triples. What happens to the force?
Scale the right side. r becomes 3r, so 1/r2 becomes 1/(3r)2 = 1/(9r2).
The force falls to one ninth of its value.
GUIDED PRACTICE
A square-root scaling
For a pendulum, T2 ∝ l. The length is quadrupled. Decide what happens to the period.
Show the working
Because T2 ∝ l, quadrupling l quadruples T2 and therefore doubles T.
The factor of four applies to T2, not to T: T changes by a factor of √4 = 2.
Where physics leans on this: Newton's law of gravitation · The inverse-square law for gamma · Pendulums and springs. All eight skills: the maths bridge.