Hooke's law — common mistakes

Always draw a clear diagram for any mechanics problem. For Hooke's Law questions, label the natural length, the stretched/compressed length, and the extension/compression. This helps avoid confusing the total length with the extension $x$ in the formula $T = \frac{\lambda x}{l_0}$.

They both measure stiffness, but in different ways. The spring constant $k$ (from $F=kx$) is the force required per unit of extension. Its units are N/m. The modulus of elasticity $\lambda$ is a property of the material and its cross-section, independent of its length. The two are related by $k = \frac{\lambda}{l_0}$, where $l_0$ is the natural length. So, a longer spring made of the same material will have a smaller spring constant $k$, but the same modulus $\lambda$.

In mechanics, 'light' is a modelling assumption that means the object has zero mass. For a light elastic string, we can ignore its weight. This simplifies calculations as we only need to consider the masses of any attached particles.

Hooke's Law, $T = \frac{\lambda x}{l_0}$, only applies within the elastic limit. Beyond this limit, the material undergoes plastic deformation, meaning it will not return to its original length when the force is removed. The relationship between tension and extension is no longer linear, and Hooke's Law is not valid.