Circular Motion

This lesson covers: 

  1. Understanding angles in radians
  2. Techniques for converting between radians and degrees
  3. Defining angular velocity (ω) as the rate of change of an angle
  4. The relationship between linear and angular velocity in rotational motion
  5. Understanding the vector nature of angular velocity
  6. Understanding frequency (f) and period (T) in circular motion
  7. The relationship between frequency (f), period (T), and angular velocity (ω)

Radians

In mathematics, we often express angles in radians. A radian is a way of measuring angles based on the length of an arc. Specifically, the angle θ in radians is defined as the ratio of the arc length to the radius (r) of the circle:


θ=rarc length​


For a full circle, which is 360°, the arc length is the same as the circle's circumference, given by 2πr. If we divide this by the radius (r), we find:


2π radians = 360°


This tells us that a full circle is equivalent to 2π radians.

Worked example - Converting degrees to radians

Convert 30° to radians.


Step 1: Conversion Formula

To convert from degrees into radians, multiply by 180π​

radians = 180∘π​×angle in degrees


Step 2: Substitute the Given Value

radians = 180π​×30=6π​

Angular velocity measures rotational speed 

Angular velocity (ω) measures the rate at which an object rotates, or changes its angle, over time. Its formula is:


 ω=ΔtΔθ​


Where:

ω = angular velocity (rad s-1)

θ = angle (rad)

t = time (s)

Linking linear and angular velocities

In circular motion, linear velocity (v) and angular velocity (ω) are related as follows:


v = r ω


Where:

v = linear velocity (m s-1)

r = radius (m)

ω = angular velocity ( rad s-1)

Worked example - Calculating angular velocity

Calculate the angular velocity of a Ferris wheel that rotates 3 times in 2 minutes.


Step 1: Formula

ω=ΔtΔθ​


Step 2: Calculate Change in Angle (Δθ)

Δθ = 3 × 2 π=6π radians


Step 3: Convert minutes to seconds (Δt)

To convert from minutes to seconds, multiply by 60

2 minutes = 120 seconds


Step 4: Substitution and correct evaluation

ω=1206π​ = 0.157 rad s−1

Frequency and period definitions

The frequency of rotation and the time period of rotation are related as follows:


f = T1​


Where:

f = number of complete rotations per second (Hz)

T = time taken for one complete rotation (s)

Relating f, T and ω

For each revolution, an object rotates through an angle of 2π radians.


Hence:

  • f revolutions per second equals ω radians per second
  • ω = 2πf
  • ω=T2π​

Worked example - Relating frequency, period, and angular velocity

Determine the angular velocity of a ceiling fan completing 120 revolutions per minute.


Step 1: Convert rpm to Frequency (f)

to convert from rpm to frequency, divide by 60

120 rpm = 2 rev s-1


Step 2: Formula

ω = 2 π× f


Step 3: Substitution and correct evaluation

ω = 2 π× 2 = 4 π rad s−1