Limit of Proportionality

The limit of proportionality is the point beyond which the extension of a spring (or wire) is no longer directly proportional to the applied force. Hooke’s law (F = kx) only applies up to this point.

On a force-extension graph

A force-extension graph for a spring starts as a straight line through the origin. This is the Hooke’s law region where extension is directly proportional to force. The limit of proportionality is the point where the graph begins to curve away from the straight line.

Beyond this point:

• The spring still stretches, but not in proportion to the force.
• The spring constant (k) no longer applies.
• The relationship between force and extension is no longer linear.

Limit of proportionality vs elastic limit

These are two different points:

The limit of proportionality is where the graph stops being a straight line.

The elastic limit is the maximum extension from which the spring can still return to its original length.

The elastic limit is at or slightly beyond the limit of proportionality. Between the two points, the spring returns to its original length but the force and extension are not proportional. Beyond the elastic limit, the deformation is permanent (plastic).

Why this matters in calculations

When an exam question gives you a spring constant and asks you to calculate extension, the answer is only valid if the force does not exceed the limit of proportionality. If the question states that Hooke’s law applies, you can use F = kx directly. If the question involves forces beyond this limit, the spring constant cannot be used.