Correction to Period of a Pendulum

Question

In one derivation of the corrected period of a pendulum, we started off like so:

The mass has a height $y$ given by $l(1-\cos \theta )$. $E = K + E \rightarrow \frac{1}{2}ml^2 \dot{\theta}^2 + mgl(1-\cos \theta)$

The next step introduces $\theta _0$, and I've got no idea where this came from.

$$\frac{1}{2}ml^2 \dot{\theta}^2 + mgl(1-\cos \theta)= mgl(1-\cos \theta _0)$$

Now we just solve for $\dot{\theta}$ and solve the DE.W I'm interested in the theta side of the equation.

$$\int \frac{d\theta}{\sqrt{\cos \theta - \cos \theta _0 }}$$

We go through a bunch of subs and changes of vairble to arrive at

$$\int ^{2\pi} _0 \frac{du}{\sqrt{1-K^2 \sin ^2 u }}$$

So my two questions are

• Why are we involving two $\theta$ values? The text didn't make it clear why we needed an extra $\theta _0$. It appears that $mgl(1-\cos \theta _0)$ is the total energy of the system. Our total energy cannot surpass the initial gravitational potential energy, this is clear. My only thought as to what $\theta$ means is the instantaneous position of the angle.
• My text mentioned that this is an elliptical integral. Mathematically speaking, what is an elliptical integral? Can this integral be solved exactly, or does it always require approximations from the expansion?

Answer 1

Since the total mechanical energy is conserved, from which we get $KE_i+PE_i=KE_f+PE_f$, it should be more clear now that $\theta_0$ is the angle at which you raise the pendulum (i.e., that term represents the $PE_i$). The terms on the left side your first centered equation are the final kinetic and potential energies, after some time $t$. In effect, you are right that $mgl\left(1-\cos\theta_0\right)$ is the total energy of the system.

An elliptic integral is one in which the function you are integrating, $f$, depends on the the variable and it's square root: $f\to f\left(x,\,\sqrt{x}\right)$. Since your function is $$ f(u,\sqrt{u})=\frac{1}{\sqrt{1-k^2\sin^2u}} $$ This is an elliptic integral. Wikipedia's article on elliptic integrals is pretty good too (and even includes your integral as an example of an elliptic integral).