Hooke's law states that the restoring force exerted by an ideal spring is directly proportional to, and opposite in direction to, its displacement from equilibrium: (F_H=-kx). It applies while the spring remains within its limit of proportionality. This is SL and HL content in A.2 Forces and momentum.
The Reasoning and Mechanism
In (F_H=-kx), (F_H) is the elastic restoring force in newtons (N), (x) is displacement from equilibrium in metres (m), and (k) is the spring constant in newtons per metre (N m(^{-1})). A larger (k) means a stiffer spring because more force is required for the same displacement.
The minus sign shows that the restoring force acts in the opposite direction to displacement. If a spring is stretched to the right, its restoring force acts to the left; if compressed to the left, the force acts to the right.
For example, a spring with (k=120\ \text{N m}^{-1}) is stretched by (5.0\ \text{cm}=0.050\ \text{m}). The magnitude of its restoring force is
Therefore, the spring exerts a (6.0\ \text{N}) force opposite to the displacement. An external force holding it stationary would have the same magnitude but the opposite direction.
A common misconception is that Hooke's law applies to every extension. Beyond the limit of proportionality, force is no longer proportional to extension; beyond the elastic limit, the spring may also remain permanently deformed.
Exam Technique
For a calculate question, convert displacement to metres, select (F_H=-kx), substitute clearly, and distinguish force magnitude from direction. On graph questions, determine (k) from the gradient only over the straight-line proportional region.
