Recipe

Apply Hooke's law to a spring problem

Use when: you're given two of the three quantities in $F=-kx$ (the spring constant $k$ , the displacement $x$ , or the restoring force $F$ ) and asked for the third

Goal: the missing quantity in $F=-kx$ , with its sign correctly read as a real physical direction

Taught in Mass-spring systems and Hooke's law

Quick recipe

Identify what's given and what's missing
Assign the correct sign to the displacement
Substitute into $F=-kx$ and solve for the unknown
Interpret the sign of the result

1
A Hooke's-law problem always hands you two of $F$ , $k$ , $x$ and asks for the third. Sort the two knowns from the one unknown before doing any algebra.

Example

A spring with $k=20$ N/m is compressed $0.4$ m; find $F$ . Here $k$ and $x$ are given, $F$ is the unknown.
- $F$ is the unknown — solve $F=-kx$ directly once $x$ 's sign is settled (Step 3).
- $k$ is the unknown — the same equation, solved for $k=-F/x$ once $x$ 's sign is settled.
Why: knowing which letter you're solving for determines which way you rearrange $F=-kx$ in Step 3, and rearranging before sorting knowns from unknowns is how sign errors creep in.
2
Stretching the spring past its natural length makes $x$ positive; compressing it makes $x$ negative. This sign is data taken from the problem, not something Step 3 solves for.

Example

Compressed $0.4$ m means $x=-0.4$ m, not $+0.4$ .
- spring stretched — $x>0$
- spring compressed — $x<0$
Why: Hooke's law's minus sign only produces a force in the physically correct direction if $x$ 's own sign already records which way the spring moved.
3
Plug the two known numbers (with $x$ 's sign already fixed) into $F=-kx$ , then solve algebraically for whichever one is missing.

Example

$F=-kx=-(20)(-0.4)=8$ . If $k$ had been the unknown instead, with $F=-18$ N given at a stretch of $x=0.6$ m, the same equation solved the other way gives $k=-F/x=-(-18)/0.6=30$ N/m.

Why: this is just isolating the one missing letter in a linear equation; the physics was already finished once $F=-kx$ was written down.
4
A positive answer for $F$ pushes toward the positive- $x$ direction; a negative one pulls toward the negative- $x$ direction. Either way, it always points back toward equilibrium, never away from it.

Example

$F=8$ N is positive: because $x$ was negative (compressed), the restoring force pushes back in the positive direction, exactly toward equilibrium.

Why: a bare number with no direction isn't actually an answer to a physics problem. Hooke's law's whole content is that the force opposes the displacement, and the sign is what shows that on paper.

Common mistakes

Writing $x$ as positive when the spring is compressed (or negative when stretched): the direction is data, not a free choice.
Dropping Hooke's law's own minus sign when solving for $k$ : $k=-F/x$ , not $F/x$ .
Reporting only a magnitude, with no direction, as the final answer.
Plugging all three given numbers into $F=-kx$ without first checking which one is actually the unknown.