Uniform Circular Motion

Uniform circular motion. The moving object travels in a circular path. The magnitude of linear velocity, acceleration, and centripetal force stays constant, but their direction changes. The red arrow represents the direction of the linear velocity and the black arrow represents the direction of the acceleration and resultant force.
Useful equations:
$$ \begin{equation} F_c = {m v^2 \over r} v = \omega r \end{equation} $$

Maximum friction force. An object is resting on a spinning surface. In this scenario, the centripetal force is the force of friction holding the object to the surface.
Useful equations:
$$ \begin{equation} F_f = \mu F_N = \mu m g \\ F_c = {m v^2 \over r} \end{equation} $$

Angled string tension. An object is connected to a point above it by a string, similarly to a tetherball. The object moves around the post in uniform circular motion, resulating in a tension force being exerted by the string at an angle \(\theta\). The horizontal component of the tension force \(F_{Tx}\) is equal to the centripetal force and the vertical component \(F_{Ty}\) is equal to \(F_g\).
Useful equations:
$$ \begin{equation} F_{Tx} = {mv^2 \over r} \\ F_{Ty} = mg \\ F_T = \sqrt{{F_{Tx}}^2 + {F_{Ty}}^2} \\ \theta = {\tan}^{-1}({F_{Tx} \over F_{Ty}}) \end{equation} $$

Uniform Circular Motion Equations
$$ \begin{equation} a_c = {v^2 \over r} = \omega^2 r \\ F_c = m a_c = {m v^2 \over r} \\[16pt] F_f = \mu F_N \\ \omega = {2 \pi \over T} \\ v = \omega r \\[16pt] T = {1 \over f} \\ f = {1 \over T} \\ \end{equation} $$