Uniform Circular Motion (UCM) - Key Parameters of Circular Motion

This is the time it takes for the object to complete one full lap (one revolution). Its unit is seconds (s).

This is the familiar speed (distance/time). In one period (T), the distance travelled is the circumference of the circle, 2πr.

v = \[\frac {Distance}{Time}\] = \[\frac {2πr}{T}\] (Unit: m/s)

It is a vector that points from the center of the circle (the origin) out to the position of the particle.

• Magnitude: Its length is simply the radius, r.
• Key Insight: As the object moves in UCM, its radius vector sweeps out equal angles in equal amounts of time.

Angular speed (ω) is the angle described by the radius vector per unit time.

ω = \[\frac {Angle Swept}{Time}\]  (Unit: radian/s)

Think of a clock's second hand. Its Period (T) is 60 seconds. The angle it sweeps in that time is 2π radians (360^°).

Calculation: During one full revolution, the angle is 2π and the time is T.

ω = \[\frac {2π}{T}\] (Unit: rad/s)

We can combine the equations for v and ω to find the relationship between the linear speed and the angular speed:

1. We know v = \[\frac {2πr}{T}\] .

2. We know ω = \[\frac {2π}{T}\] .

By substituting the term 2π/T from the ω equation into the v equation:

On a merry-go-round, you and a friend near the center have the same angular speed (ω) (you both complete a circle in the same time, T). However, you (on the outer edge with a larger r) have a much faster linear speed (v) because you travel a greater distance in the same amount of time!