6. Circular Orbits

Derivations

Cetripetal acceleration must equal force due to gravity

$$F_c = m\omega ^2 r, \quad \quad F_g = \dfrac{GMm}{r^2}$$

equating these

$$m\omega^2 r = \dfrac{GMm}{r^2}$$

simplify

$$\omega^2 = \dfrac{GM}{r^3}$$

 

Angular Velocity

$$\omega = \sqrt{\dfrac{GM}{r^3}}$$

 

Linear Velocity

we sub $\omega = \dfrac{v}{r}$

$$\dfrac{v^2}{r^2} = \dfrac{GM}{r^3}$$

simplify

$$v = \sqrt{\dfrac{GM}{r}}$$

 

Useful Equations

$$\omega = \sqrt{\dfrac{GM}{r^3}}, \quad \quad v = \sqrt{\dfrac{GM}{r}}$$