Hooke’s law connects how much an ideal spring is stretched or compressed to the force it exerts. In AP Physics 1, it provides a simple linear model for spring forces in many translational dynamics situations.

What Hooke’s Law States

Hooke’s law says the magnitude of the spring force increases in direct proportion to the spring’s displacement from its relaxed (unstretched) length, as long as the spring remains in its linear range.

A larger spring constant means a stiffer spring: it produces more force for the same displacement.

Hooke’s law is often written with a negative sign, $F_x=-kx$, to show direction, but the syllabus emphasis here is on the magnitude relationship.

Displacement, Direction, and Sign Conventions

In 1D problems, you choose a positive axis and then assign a sign to the spring force using your coordinate choice.

• If you use $F_x=-kx$, then:

  • $x$ is a signed displacement from equilibrium.

  • The negative sign encodes that the spring force points opposite the displacement (a restoring tendency).

• If you use $F_s=kx$, then:

  • $x$ is treated as a magnitude, and you assign the force direction separately based on whether the spring is stretched or compressed.

Graphical Meaning (Force–Displacement)

Hooke’s law implies a straight-line relationship between spring-force magnitude and displacement magnitude.

Force magnitude vs. displacement for an ideal Hooke’s-law spring forms a straight line through the origin, so the proportionality constant is the slope: $k=\Delta F/\Delta x$. The accompanying hanging-mass setup illustrates how measured weight (force) and extension data generate the linear plot used to determine $k$. Source

• A graph of $F_s$ vs. $x$ is linear and passes through the origin (ideal behaviour).

• The slope of the line is $k$ (units N/m), so you can determine $k$ experimentally from force and displacement data.

• If the graph noticeably curves, the spring is no longer behaving ideally; Hooke’s law is not a good quantitative model there.

Using Hooke’s Law in Dynamics Setups

When a spring is part of a force analysis, the spring force acts along the spring and should be treated like any other external force on the chosen object.

• Identify the object whose forces you are summing.

• Determine the spring’s displacement $x$ from its relaxed length (watch units: cm to m).

• Use $F_s=kx$ for magnitude, then apply the correct direction in your free-body reasoning.

• Combine with Newton’s second law algebraically when needed: the spring force may oppose or reinforce other forces depending on geometry and displacement direction.

The key modelling step is that the spring force depends on how far the spring is displaced, not on the object’s mass or speed.

Limits and Common Pitfalls

Hooke’s law is a model with a restricted domain.

• Valid only within the spring’s linear (elastic) range; beyond that, $F$ is not proportional to $x$ and permanent deformation can occur.

• $x$ is not the spring’s total length; it is the change from relaxed length.

• Confusing direction: $k$ is always positive; the sign comes from your axis choice, not from $k$.

• Using the wrong units: $k$ in N/m requires $x$ in metres.