Formulas in Uniform Circular Motion

• Magnitude of instantaneous acceleration: \(|a|=\displaystyle\lim_{\Delta t \rightarrow 0} \dfrac{|\Delta v|}{\Delta t}\) = \(\dfrac{dv}{dt}\)

• Angular speed \(\omega=\dfrac{\Delta \theta}{\Delta t}\)

• Linear speed \(v=R\omega\)

• Centripetal acceleration \((a_c)\) : \(a_c=\dfrac{v^2}{R}=\omega^2 R\) directed towards center of path followed by object

• Centripetal force \((F_c)\) : \(F_c=m a_c=m \omega^2 R=m \dfrac{v^2}{R}\) directed towards center of path followed by object

Inertia keeps objects from falling into the center of the path they follow, while the centripetal force keeps them from flying off.