Summary

General Principles

Expressed in x- and y-component form:

\begin{array}{l} {(F_{net})}_{x} = \sum^{​} F_{x} = m a_{x} \\ {(F_{net})}_{y} = \sum^{​} F_{y} = m a_{y} \end{array}

Expressed in rtz-component form:

\begin{array}{l} {(F_{net})}_{r} & = & \sum^{​} F_{r} = m a_{r} = \frac{m v_{t}^{2}}{r} = m ω^{2} r \\ {(F_{net})}_{t} & = & \sum^{​} F_{t} = {\begin{array}{l} 0 & uniform circular motion \\ m a_{t} & nonuniform circular motion \end{array} \\ {(F_{net})}_{z} & = & \sum^{​} F_{z} = 0 \end{array}

Speed is constant.
$\vec{F}_{net}$ points toward the center of the circle.
The centripetal acceleration $\vec{a}$ points toward the center of the circle. It changes the particle’s direction but not its speed.

With no drag, the x- and y-components of acceleration are independent. The trajectory is a parabola.
With drag, the trajectory is not a parabola. Maximum range is achieved for an angle less than $45 °$ .

An object acted on only by gravity has a circular orbit of radius r if its speed is

v = \sqrt{r g}

The object is in free fall.

Circular motion requires a net force pointing to the center. n must be $> 0$ for the object to be in contact with a surface.