# UNIFORM CIRCULAR MOTION

Uniform circular motion is a motion where an object moves in a circle at constant speed v. The radius of the circle is r and the velocity vector $$\vec { v }$$ is perpendicular to position vector and tangential to the circle. The velocity vector $$\vec { v }$$ is continuously changing but the speed is constant.

The period T is the time required for one complete revolution of the object around the circle. The unit of period T is seconds. Frequency f is the number of rotations per second. The unit for frequency is sec-1 or  “hertz”. Period and frequency are related by following equation.

$$f=\frac { 1 }{ T }$$ (Eq-1)

Angular velocity ω is the rate of change of angular displacement. Since there are 2π radians in one full circle and it takes T seconds to go around once, it’s obvious that omega equals 2π divided by T.

$$\omega =\frac { 2\pi }{ T }$$ (Eq-2)

The speed v is the circumference of circle divided by the time to go around once (T).

$$v=\frac { 2\pi r }{ T }$$ (Eq-3)

Since ω=2π/T, Eq-3 can be rewritten as

$$v=wr$$. (Eq-4)

As stated earlier, the velocity vector is continuously changing so there must be acceleration. The acceleration that changes the velocity vector is always pointing towards the center of the circle and it’s called centripetal acceleration. Centripetal acceleration is a vector quantity and always perpendicular to $$\vec { v }$$.

The magnitude of the centripetal acceleration is

$$\left| { a }_{ c } \right| =\frac { { v }^{ 2 } }{ r } ={ w }^{ 2 }r$$. (Eq-5)