Pendulum speed at angle $\phi $, expressing $d\vec{l}$ in terms of angle [closed]

Question

Problem statement: Pendulum swings at angle $\phi = \phi_{0}$ , and $v_{\phi_0} = 0$. We want to find the velocity when it swings at an angle $\phi$.

Attempt at problem: By work energy theorem $$ \frac{1}{2}mv_{\phi}^{2} - \frac{1}{2}mv_{\phi_0}^{2}= \int \vec{F} \ \cdot \ d\vec{l} $$

Since $T \perp d\vec{l} \implies T = \vec{0}$.

So, $F (\hat{r})= -mg\cos(\phi-\frac{\pi}{2})\hat{r}$ is the force acting on the mass during the swing, respectively to $\hat{r}$ direction.

We then get $$W = \int_{\phi_0}^{\phi} -mg\cos(\phi-\frac{\pi}{2})\ d\vec{l}$$

But here I must express $d\vec{l}$ in terms of $d\phi \ $ and this is where I'm lost. According to my textbook, $d\vec{l} = l\ d\phi$ which I can't understand how and why!

Answer 1

$l$ must mean the length of the string, which is the radius of the circle the mass moves in. Arc length is given by $s=r\phi$ so the differential change in displacement is given by $ds=rd\phi$, where $r=l$.