Circular Motion, SHM & Oscillations
The S.I. unit for angles is not degrees, but radians. This is the angle subtended by a circular arc with a length equal to the radius of the circle:
1 rad is equal to about 57.3°
There are 2π in a circle, and half that in a semicircle. Therefore,
to convert from degrees to radians, divide the angle in degrees by 180/π
to convert from radians to degrees, multiply the angle in radians by 180/π
While the velocity of linear motion is defined as displacement / time, the velocity of circular motion is defined as angle / time. Therefore, it is called angular velocity, and is given the Greek letter omega, ω.
ω = θ/t angular velocity = angle / time
This equation is rarely used, however, as it is far easier to work with the time period and frequency of circular motion.
Time period, T, is the time taken for it to complete one circle.
Frequency, f, refers to the number of complete rotations (of 2π rads) per second.
This gives rise to the far more helpful equations:
ω = 2π/T angular velocity = 2π / time period
ω = 2πf angular velocity = 2π x frequency
The units of angular velocity are radians per second.
We can calculate the linear velocity using the angular velocity:
v = ωr linear velocity = angular velocity x radius
Even if the object undergoing circular motion is travelling at a constant speed, its velocity is always changing (due to its direction changing). This means the object is always accelerating – this is known as centripetal acceleration, and always acts towards the centre of the circle.
a = v²/r centripetal acceleration = linear velocity² / radius
a = ω²r centripetal acceleration = angular velocity² x radius