# Simple Pendulum Why Generalized Coordinate Always Angle?

The force of gravity is in the $\hat y$ direction, but that's not the only force in the problem. There's also tension in the string, which points along the string. The *total* force then points tangentially to the circle.

Another way to say this is: You can work in coordinates $x,y$ with basis vectors $\hat x$, $\hat y$, but then the force points in both the $\hat x$ and $\hat y$ directions. If you instead work in the coordinates $r,\theta$, with unit vectors $\hat{r}$, $\hat{\theta}$, you'll find that the force points only in the $\hat \theta$ direction, with no component along $\hat r$. So these are nicer coordinates to use!

Since the string is of fixed length l, we can write $x=\sqrt(l^2-y^2)$

Umm no

$x=\pm\sqrt{l^2-y^2}$

So there are two different positions of the system with the same $y$ value, one with positive $x$ and one with negative $x$, a little bit of thought about how a pendulum swings shows that both positive and negative $x$ values are part of the normal operating region of the pendulum.

You can use any coordinate system you like. Some of them make it a lot easier to solve the equations of motion however. In particular if you choose $\theta$ then you end up with a system which manifestly has one degree of freedom, while if you choose $x$ & $y$ you need to express it as being in two dimensions with a constraint between them: $y = -\sqrt{l^2 - x^2}$.

People generally like to choose the coordinate system which makes the solution easiest.