newtonian mechanics - Why is the simple harmonic motion idealization inaccurate?

While in my physics classes, I've always heard that the simple harmonic motion formulas are inaccurate e.g. In a pendulum, we should use them only when the angles are small; in springs, only when the change of space is small. As far as I know, SHM came from the differential equations of Hooke's law - so, using calculus, it should be really accurate. But why it isn't?

Answer

The actual restoring force in a simple pendulum is not proportional to the angle, but to the sine of the angle (i.e. angular acceleration is equal to $-\frac{g\sin(\theta)}{l}$, not $-\frac{g~\theta}{l}$ ). The actual solution to the differential equation for the pendulum is

$$\theta (t)= 2\ \mathrm{am}\left(\frac{\sqrt{2 g+l c_1} \left(t+c_2\right)}{2 \sqrt{l}}\bigg|\frac{4g}{2 g+l c_1}\right)$$

Where $c_1$ is the initial angular velocity and $c_2$ is the initial angle. The term following the vertical line is the parameter of the Jacobi amplitude function $\mathrm{am}$, which is a kind of elliptic integral.

This is quite different from the customary simplified solution

$$\theta(t)=c_1\cos\left(\sqrt{\frac{g}{l}}t+\delta\right)$$

The small angle approximation is only valid to a first order approximation (by Taylor expansion $\sin(\theta)=\theta-\frac{\theta^3}{3!} + O(\theta^5)$).

And Hooke's Law itself is inaccurate for large displacements of a spring, which can cause the spring to break or bend.