Spring Constant

The spring constant is a property of the spring itself, in the same way that mass is a property of a block. It is the one number you have to be told, or measure, before any spring problem has an answer.

Its unit is newtons per meter. That falls straight out of $\vec{F}_s = -k\Delta\vec{x}$: a force divided by a length. A spring with $k = 250\ \text{N/m}$ takes $250\ \text{N}$ to stretch by one metre, or $2.5\ \text{N}$ to stretch by one centimetre. The AP Physics 1 Table of Information lists $k$ as the spring constant in both symbol columns of its Mechanics and Fluids panel.

Three printed equations contain it. On the AP Physics 1 equation sheet those are $\vec{F}_s = -k\Delta\vec{x}$, the stored energy $U_s = \frac{1}{2}k(\Delta x)^2$, and the oscillation period $T_s = 2\pi\sqrt{m/k}$. The CED adds one more that the sheet does not print: essential knowledge 7.4.A.4.ii gives the total energy of a spring and object system as $E_{\text{total}} = \frac{1}{2}kA^2$.

You measure it as a slope. Stretch the spring by known amounts, plot applied force against stretch, and $k$ is the gradient of the straight line. Topic 2.8 lists skill 1.B, creating quantitative graphs with appropriate scales and units, among its suggested skills, which is that experiment.

It does not change when the situation does. Hanging a heavier block, pulling further, or oscillating faster leaves $k$ alone; only cutting or replacing the spring changes it. The relationship $k$ sits inside is Hooke's law.