Angular frequency

In physics, angular frequency, ω, (also called the angular speed, radial frequency, and radian frequency) is a measure of rotation rate. A high rate of angular frequency means something is turning very fast. The angular frequency is the magnitude of the vector quantity angular velocity which is also known as angular frequency vector . 
It is useful in many areas of maths and science as it allows understanding of many properties of physical objects in our world. In SI units, angular frequency is measured in radians per second, with dimensions t−1 since radians are dimensionless. 

 Theory 
Angular frequency is often given in radians per second ()  as it is easier to work with. In this way, the angular frequency is given by, 



where  is the time (period) of a single rotation (revolution) and  is the frequency.  This can be derived by considering  when  and .

If a wheel turns by an angle  in a time  then the angular frequency at any moment is given by, 



If the rotation speed is constant, this becomes, 



and so it is similar to a speed.   

For objects which are going in circles, the angular frequency may also be given as (in radians), 



where  is the speed and  is the distance from the center of the circle.

This can be derived by considering  where  is the distance along the arc.

Thus 



where  is the velocity.

 Examples 
The angular frequency is important in determining whether an object can stay above the ground against gravity, or whether a spinning top can stay standing. It also is important in creating the frequency of mains electricity supplies and reducing the heat due to friction in engines. 

 Satellites 
Objects are pulled towards the earth because of gravity. To oppose this, a satellite must go fast enough it misses the ground. For the forces to equal each other, 



where  is the mass of the larger body and  is the Gravitational constant. To make a circle, the angular frequency must be equal to, 



which leads to one of Johannes Kepler's Laws.

 Mains Supplies 
Generators in power stations rotate at some frequency.  The speed at which they turn gives us the utility frequency of our electricity supply.  These can be slowed by large brakes or big heavy wheels which stop the generators from turning too quickly.

 Units 

 Measurements 
NameSymbolDefinitionRadians per second Revolutions per minute Frequency per secondDegrees per second

Related pages
Frequency
Radian
Angular velocity
Angular momentum

Category:Physical quantity
Category:Mechanics
Category:Basic physics ideas