# 9.1: Introduction Newton’s Second Law and Circular Motion

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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,

$F_{\theta}=0 \nonumber$

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