# Why is a pendulum an example of harmonic motion?

I've heard a lot of people say that a pendulum moving back and forth on a fixed-length spring is an example of harmonic motion. But when I derive the governing equation for a pendulum, I get

$$\theta'' = \frac{-g}{L} \sin(\theta)$$

(see for example video here for a full derivation)

and I thought the equation of a harmonic oscillator was

$$x'' = -k x$$

like for springs. Why do people call a pendulum an example of harmonic motion if it doesn't satisfy the second equation?

(I think it's just because they use the small angle approximation $$\sin(\theta) \approx \theta$$ and call it a day.)

• You answered yourself with the last sentence in parentheses. – wcc Apr 2 at 14:54
• Yeah, it seems like you figured this one out. Pendulum equations only apply for small perturbations, and even then it's only approximate. – JMac Apr 2 at 14:58
• Note that Hooke's law is only approximately true so even for a spring the equation is actually $\ddot x = -kx + O(x^2)$ – John Rennie Apr 2 at 15:00
• I'll "connect the final dot". sin($\theta$) = $\theta$. Acceleration of the pendulum bob is proportional to $\theta$ because $a=g sin(\theta)$, so the force on the pendulum bob is also proportional to $\theta$. For any object whose restoring force is linear and opposes the displacement from equilibrium, that object demonstrates simple harmonic motion – David White Apr 2 at 15:02
• I originally overthought this and tried to show $\sin(\theta)'' = -k \sin(\theta)$ for some $k$. I was very unsuccessful. – overseas Apr 3 at 14:20