In physics, a rigid body is an object with a definite shape and size that does not deform under external forces. When such a body rotates about a fixed axis, every point on the object moves in a circular path around that axis. The study of circular motion involves concepts analogous to those in linear motion but adapted for circular paths.

Angular Position (\(\theta\))

• In linear motion, position is described by coordinates (e.g., x, y).
• Meanwhile, in circular motion, angular position (\(\theta\)) specifies the orientation of the body relative to a reference axis.
• It is measured in radians (rad) where:
  • s: Arc length
  • r: Radius of the circular path
  

Angular Velocity (\(\omega\))

• Analogous to linear velocity (v), angular velocity (\(\omega\)) describes how fast the angle changes with time.
• It is measured usually in rad/s or rpm (revolutions per minute), where \(1 \; \text{rpm} = \frac{\pi}{30} \; \text{rad/s}\)
• Average angular velocity = \(\omega _{avg} = \frac{\Delta \theta}{\Delta t}\)

Angular Acceleration (\(\alpha\))

• Analogous to linear acceleration (a), angular acceleration (\(\alpha\)) measures the rate of change of angular velocity.
• It is measured in \(\text{rad/s}^2\).
• Average angular acceleration = \(\alpha _{avg} = \frac{\Delta \omega}{\Delta t}\)

Relating Linear and Angular Quantities

For a point on a rotating rigid body at a distance r from the axis of rotation

Velocity (v vs \(\omega\))

• The tangential speed of the point is \(v_{tan} = r \omega\)
• The farther the point is from the axis, the greater its linear speed for the same \(\omega).
• Note: the total velocity of a point in a spinning object is the vectorial sum of the tangential velocity and the center-of-mass velocity \(v_p = v_{cm} + v_{tan}\)

Acceleration (a vs \(\alpha\))

• Tangential acceleration (changes speed) = \(a_{tan} = r \alpha \)
• Centripetal acceleration (changes direction, points toward the center) = \(a_{rad} = \frac{v^2}{r} = \omega ^2 r\)
• Learn more about this in centripetal force
• The total linear acceleration is the vector sum of tangential and centripetal components.

Period and Frequency

Period (T)

• Time it takes for a rotating object to complete one full revolution (360° or 2π radians).
• Measured in seconds (s).
• \(T = \frac{2 \pi}{\omega} = \frac{1}{f}\)

Frequency (f)

• Number of complete revolutions per unit time.
• Measured in Hertz (Hz)
• \(f = \frac{1}{T} = \frac{\omega}{2 \pi}\)

Now that we have established the basic concepts, let's look at some examples.

Example 1: Bicycle Wheel Example 2: Rotating Disc Example 3: Car Tyre

Written by Mateo Sancho