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 = r θ

Let r = A = radius of the circle.

The angular velocity (ω) is given by:

ω = \( \frac{θ}{t} = \frac{s}{r} \: \times \: \frac{1}{t} \)

= \( \frac{s}{t} \times \frac{1}{r} \)

\( \frac{s}{t} \)= v, the linear velocity of the particle.

Therefore;

ω = v. \( \frac{1}{r}\)

V = ωr = ωA.

Linear speed is the product of the angular speed and the radius or amplitude of motion.

Where,

v is