A student stretches a spring by hanging masses from it. The student notices that when the load is small, doubling the load doubles the extension. However, beyond a certain point, doubling the load produces an extension more than double the previous extension. Explain these observations, using the terms limit of proportionality and spring constant in your answer.

Below the limit of proportionality the spring obeys Hooke’s law: the extension is directly proportional to the applied force, so doubling the load doubles the extension. The spring constant, k, remains constant in this linear region.

Beyond the limit of proportionality the spring no longer follows Hooke’s law. The extension is no longer directly proportional to the load; a smaller additional force is required to produce each further unit of extension. Consequently, when the load is doubled in this non‑linear region the extension increases by more than twice the previous value.