Relationship between Angular Acceleration & Linear Acceleration

Speed:

From the diagram, as the particle P moves round the circle once it sweeps through an angle = 360º. Where 360º = 2π radians, in time T seconds.

The time rate of change of angle with time (t) is called the angular velocity (ω).   

ω = \( \frac{angle \: turned \: through \: the \: body}{time \: taken} \)

ω = \( \frac{θ}{t}\)

∴ θ = ωt

This is similar to:
distance = velocity × time

s = v × t

∴ v = \( \frac{s}{t}\) for linear motion;

The angle θ is measured in radians
since, 2π rad = 360º
from, ω = \( \frac{θ}{t}\)
angular velocity is measured in radians per second (rad/s)

When θ changes with time, the length of the arc PZ = s, also changes with time.

By definition, θ in radians = \( \frac{s}{r}\)

s = rθ

Let r = A = radius of the circle.

The angular velocity (ω) is given by: