Angular Speed

Any object travelling in a uniform circular motion at the same speed travels with a constantly changing velocity
This is because it is constantly changing direction, and is therefore accelerating
The angular speed (⍵) of a body in circular motion is defined as:

The rate of change in angular displacement with respect to time

Angular speed diagram, downloadable AS & A Level Physics revision notes

When an object is in uniform circular motion, velocity constantly changes direction, but the speed stays the same

Calculating Angular Speed

Taking the angular displacement of a complete cycle as 2π, the angular speed ⍵ can be calculated using the equation:

$ω = \frac{∆ θ}{∆ t} = \frac{2 π}{T} = 2 π f$

Where:
- Δθ = change in angular displacement (radians )
- Δt = time interval (s)
- T = the time period (s)
- f = frequency (Hz)
Angular velocity is the same as angular speed, but it is a vector quantity
When an object travels at constant linear speed v in a circle of radius r, the angular velocity is equal to:

$ω = \frac{∆ θ}{∆ t} = \frac{v}{r}$

Where:
- v is the linear speed (m s^-1)
- r is the radius of orbit (m)
This equation tells us:
- The greater the rotation angle θ in a given amount of time, the greater the angular velocity ⍵
- An object rotating further from the centre of the circle (larger r) moves with a smaller angular velocity (smaller ⍵)

A bird flies in a horizontal circle with an angular speed of 5.25 rad s^-1 of radius 650 m.

Calculate:

a) The linear speed of the bird

b) The frequency of the bird flying in a complete circle

Answer:

Step 1: List the known quantities

Step 2: State the linear speed equation

$v = r ω$

Step 3: Calculate linear speed

$v = 650 \times 5.25 = 3410 m s^{- 1} (3 s . f .)$

Step 2: Equate angular speed with the frequency equation

$ω = 2 π f$

Step 3: Rearrange to make frequency the subject

$f = \frac{ω}{2 π}$

Step 4: Substitute the known values to calculate

$f = \frac{5.25}{2 π} = 0.836 Hz (3 s . f .)$