Circular Motion

Radians and Degrees

1 Radian = 57.30 degrees

One complete Radian 360^θ = 2π

(Value) x π/180 = Radians

e.g. 60^θ x π/180 = 1.05 rad

(Value) x 180/π = Degrees

e.g. 3 rad x 180/π = 171.89^θ

Angular velocity is a measure of the rate of rotation of an object around a specific axis. It is defined as the angle rotated per unit time.

Unit: radians per second (rad/s).

ω = ΔΘ/Δt

ω = 2π/T

ω = 2πf

Linear velocity is the rate at which an object moves along a straight path. It is defined as the displacement of the object per unit of time.

Unit: meters per second (m/s).

v =d/t

v = 2rπ / T

T = 2π / ω

V = rω

Arc length is the distance along a curved line or the portion of the circumference of a circle between two points.

s = 2𝛑r x Θ / 2𝛑 = rΘ

Angular frequency, often denoted by ω (omega), is a measure of how quickly an object oscillates or rotates in radians per second.

ω = 2𝛑 / T = 2𝛑f

The centripetal force is the resultant force acting on a body moving at constant speed in a circle.

a = v²/r = ω²r

F = mv²/r = mrω²

The forces that cause circular motion include:

Centripetal Force: The net force directed towards the center of the circular path, keeping the object in circular motion.
Frictional Force: In some cases, friction between surfaces can provide the necessary centripetal force.
Tension: In systems like a swinging pendulum or a tethered object, tension in the string or rope can act as the centripetal force.
Gravitational Force: In planetary motion, gravity acts as the centripetal force keeping planets in orbit around the sun.

Direction is always changing, so velocity is always changing
There must be a constant acceleration
This acceleration is directed to the centre of the circle
There must also be a force to the centre of the circle - this is the centripetal force

Centripetal Force + Reaction to Centripetal Force (inertia)