# 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.