Physics Course/Motion/Periodic Motion/Circular Motion

Circular Motion

θ = angular displacement
r = radius/position vector
P = position of the object along the path
v = linear velocity
Ω = angular velocity along axis of rotation
a = centripetal acceleration

Circular Motion is a motion of an object along a circular path. If the speed of the body remains constant throughout the motion, the object is said to perform a uniform circular motion. For an object in uniform circular motion along a circular path of radius R and $\overrightarrow{r}$ be the position vector of the object with the center of the path as the and $\hat{r}$ being the unit vector along it and T be the time taken to traverse the path once (period), the total linear distance covered in one period is (the circumference of the circle)

$s=2\pi R$

The speed (or linear velocity) is then given by

$v=\frac{s}{T}=\frac{2\pi R}{T}=2\pi fR\dots (f=\frac{1}{T}=\text{frequency})$

The linear velocity is a vector quantity whose direction at any given instance is tangential to the circle at that point. The angular velocity around the circle is

$\overrightarrow{\omega }=\frac{\overrightarrow{r}\times \overrightarrow{v}}{{\left|\overrightarrow{r}\right|}^2}$

Due to the vector product, the angular velocity vector is perpendicular to the plane of motion.

With circle of radius R = 1

$\omega =2\pi f$