# Physics Study Guide/Circular Motion

## Uniform Circular Motion

### Speed and frequency

!A two dimensional polar co-ordinate system. The point $M$ can be
located in 2D plane as $(a,b)$ in Cartesian coordinate
system or $(r,\theta)$ in
polar coordinate
system in Cartesian coordinate system or (r,\theta) in polar coordinate system")

Uniform circular motion assumes that an object is moving (1) in circular
motion, and (2) at constant speed $v$; then

where $r$ is the radius of the circular path, and $T$ is the time period
for one revolution.

Any object travelling on a circle will return to its original starting
point in the period of one revolution, $T$. At this point the object has
travelled a distance $2\pi r$. If $T$ is the time that it takes to
travel distance $2\pi r$ then the object\'s speed is

where $f=\frac1T$

### Angular frequency

Uniform circular motion can be explicitly described in terms of polar
coordinates through angular frequency, $\omega$ :

where $\theta$ is the angular coordinate of the object (see the diagram
on the right-hand side for reference).

Since the speed in uniform circular motion is constant, it follows that

From that fact, a number of useful relations follow:

The equations that relate how $\theta$ changes with time are analogous
to those of linear motion at constant speed. In particular,

The angle at $t=0$, $\theta_0$, is commonly referred to as *phase*.

### Velocity, centripetal acceleration and force

The position of an object in a plane can be converted from polar to
cartesian coordinates through the equations

Expressing $\theta$ as a function of time gives equations for the
cartesian coordinates as a function of time in uniform circular motion:

Differentiation with respect to time gives the components of the
velocity vector:

Velocity in circular motion is a vector tangential to the trajectory of
the object. Furthermore, even though the speed is constant the velocity
vector changes direction over time. Further differentiation leads to the
components of the acceleration (which are just the rate of change of the
velocity components):

The acceleration vector is perpendicular to the velocity and oriented
towards the centre of the circular trajectory. For that reason,
acceleration in circular motion is referred to as *centripetal
acceleration*.

The absolute value of centripetal acceleration may be readily obtained
by

For centripetal acceleration, and therefore circular motion, to be
maintained a *centripetal force* must act on the object. From Newton\'s
Second Law it follows directly that the force will be given by

the components being

and the absolute value

thumbtime=5:44\|middle\|720px\|Example of finding the centripetal
acceleration of moon in
orbit.\'\'

