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

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!

$$l$$ 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 $$r=l$$.