Why does the equation for time period of a simple pendulum become less accurate at angular displacements greater than 20 degrees?

Question

I am currently writing up a practical involving a simple pendulum and changing the weight of a fixed length to see how that affects the time period. This can be defined using the equation: $t = 2\pi \sqrt{\ell/g}$. However after some research, I discovered that this equation only works when working with angular displacements of less than 20 degrees, and at higher angles its precision falls. Can someone please explain to me why this is as I cannot find answers anywhere.

Answer 1

The bob of the pendulum will be moving back and forth in an arc of a circle. When its string (of length $l$) is at angle $\theta$ (in radians) to the vertical, the directed distance, $s$, of its bob, measured along the arc from its equilibrium position at the bottom of the arc, is $s=l \theta$.

Imagine (or better, draw a diagram of) the pendulum with its string at angle $\theta$ to the vertical. There will be two forces acting on the bob: the pull of the Earth's gravity ($mg$ downwards) and the tension force $T$ from the string. Of these, only the pull of gravity has a component along the arc. The component is of magnitude $mg \sin \theta$.

We therefore have this equation of motion for the bob's motion along the arc: $$m\ddot s=-mg \sin \theta$$ in which $\ddot s$ means the acceleration of the bob along the arc in the direction of $s$, that is away from the bottom of the arc.

We have already noted that $s=l \theta$, so, since $l$ is constant, we have $\ddot s = l \ddot \theta$, so we can write our equation of motion in terms of $\theta$ as $$\ddot \theta=-\frac gl \sin \theta$$ If, but only if, $\theta$ is small (< 20° is an arbitrary choice) we can make the approximation $\sin \theta=\theta$, so the equation of motion becomes $$\ddot \theta=-\frac gl \theta$$ which represents simple harmonic motion of period $T=2\pi \sqrt \frac lg$. [This needs a little calculus to show.] If the motion is through angles that are not small, the motion, though still periodic, will have a period that depends on the amplitude, and is not given by the famous equation just quoted!

Answer 2

The classic equation for a pendulum uses the handy fact that sin(theta) is approximately equal to theta for small values of theta. This makes the math a lot easier, but prohibits the use of large angles of swing because the approximation is no good out there.

Answer 3

The solution you cite is for the so-called small angle approximation. In this approximation in the Newtonian Equation of Motion (NEM), the sine:

$$\sin \theta \approx \theta$$

for $\theta << 1\,\mathrm{radians}$.

This condition make solving the NEM much easier, which would otherwise be:

$$\ddot{\theta}+\omega^2\sin \theta=0$$