Circular motion

Angular quantities

The hitherto used displacement / space quantities (s, h, x, y), speed (v) and acceleration (The), were useful when the objective was to describe linear movements.

In the analysis of circular motions, we must introduce new quantities, which are called angular quantities, always measured in radians. Are they:

  • displacement / angular space: φ (phi)
  • angular velocity: ω (omega)
  • angular acceleration: α (alpha)

Know more…

From the definition of radian we have:

From this definition it is possible to obtain the relation:

And it is also possible to know that the arc corresponding to 1rad is the angle formed when its arc s has the same radius length R.

Angular Space (φ)

Angular space is called the space of the arc formed when a piece of furniture is at any angle opening φ relative to the point called origin.

E is calculated by:

Angular displacement (Δφ)

As for linear displacement, we have an angular displacement if we calculate the difference between the final angular position and the initial angular position:


By convention:

Counterclockwise the angular displacement is positive.

Clockwise the angular displacement is negative.

Angular velocity (ω)

Analogous to linear velocity, we can define the average angular velocity, as the ratio of angular displacement by time interval of motion:

Your unit in the International System is: rad / s

Also found: rpm, rev / min, rev / s.

You can also set the instantaneous angular velocity as the average angular velocity limit when the time interval tends to zero:

Angular Acceleration (α)

Following the same analogy used for angular velocity, we define mean angular acceleration as:

Some important relationships

Through the definition of radian given above we have to:

but if we isolate S:

deriving this equality on both sides as a function of time we will obtain:

but the derivative of position versus time equals linear velocity and the derivative of angle position versus time equals angular velocity, thus:

where we can again derive equality as a function of time and get:

but the derivative of linear velocity versus time equals linear acceleration, which in circular motion is tangent to the path, and the derivative of angular velocity versus time equals angular acceleration, so:


Linear Angular
s = φR
v = ωR
The = αR

Period and Frequency

Period (T) is the minimum time interval for a cyclic phenomenon to recur. Your unit is the unit of time (second, minute, hour…)

Frequency(f) is the number of times a phenomenon occurs in a certain unit of time. Its most common unit is Hertz (1Hz = 1 / s) being also found kHz, MHz and rpm. In circular motion the frequency equals the number of revolutions per second and is equivalent to angular velocity.

To convert revolutions per second to rad / s:

knowing that 1 rotation = 2πrad,

Uniform Circular Motion

A body is in Uniform Curvilinear Motion if its trajectory is described by a circle with an "axis of rotation" at a distance R, and its velocity is constant, that is, the same at all points of the course.

In everyday life, we see many examples of MCU, such as a ferris wheel, a carousel or the blades of a spinning fan.

Although the linear velocity is constant, it undergoes a change of direction and direction, so there is an acceleration, but since this acceleration does not influence the velocity modulus, we call Centripetal Acceleration.

This acceleration is related to angular velocity as follows:

Knowing that is that , you can convert the hourly function from linear to angular space: