9.1: Introduction Newton’s Second Law and Circular Motion

I shall now recall to mind that the motion of the heavenly bodies is circular, since the motion appropriate to a sphere is rotation in a circle.

Nicholas Copernicus

We have already shown that when an object moves in a circular orbit of radius r with angular velocity $\overrightarrow{\boldsymbol{\omega}}$ it is most convenient to choose polar coordinates to describe the position, velocity and acceleration vectors. In particular, the acceleration vector is given by

$$\overrightarrow{\mathbf{a}}(t)=-r\left(\frac{d \theta}{d t}\right)^{2} \hat{\mathbf{r}}(t)+r \frac{d^{2} \theta}{d t^{2}} \hat{\boldsymbol{\theta}}(t) \nonumber$$

Then Newton’s Second Law, $\overrightarrow{\mathbf{F}}=m \overrightarrow{\mathbf{a}}$ can be decomposed into radial $(\hat{\mathbf{r}}-)$ and tangential $(\hat{\boldsymbol{\theta}}-)$ components

$$F_{r}=-m r\left(\frac{d \theta}{d t}\right)^{2}(\text { circular motion }) \nonumber$$

$$F_{\theta}=m r \frac{d^{2} \theta}{d t^{2}} \quad(\text { circular motion }) \nonumber$$

For the special case of uniform circular motion, $d^{2} \theta / d t^{2}=0$, and so the sum of the tangential components of the force acting on the object must therefore be zero,

$$F_{\theta}=0 \nonumber$$