Physics Course/Motion/Periodic Motion/Circular Motion

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==Circular Motion==,br>

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 \vec{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\qquad\qquad\ldots\left(f=\frac{1}{T}\,=\textrm{frequency}\right)

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

\vec{\omega} = \frac{\vec{r}\times\vec{v}}{\left|\vec{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