Angular velocity, unlike linear velocity, does not change with radius.
Understanding Angular Velocity
Angular velocity (ω) is a measure of how fast an object rotates or revolves around a central point. It describes the rate at which the angle changes over time.
As stated in the reference:
Whereas the linear velocity measures how the arc length changes over time, the angular velocity is a measure of how fast the central angle is changing over time. ω=θt. The symbol ω is the lower case Greek letter “omega.” Also, notice that the angular velocity does not depend on the radius r.
This fundamental definition highlights that angular velocity is focused purely on the rotational speed in terms of angle per unit time (e.g., radians per second, degrees per minute).
Angular Velocity vs. Linear Velocity
It's important to distinguish angular velocity from linear velocity (v).
- Angular Velocity (ω): Measures rotational speed (how fast the angle changes). It is independent of the radius.
- Linear Velocity (v): Measures tangential speed (how fast a point on the rotating object travels along its path). It is directly dependent on the radius.
The relationship between the two is given by:
v = rω
This equation shows that for a fixed angular velocity (ω), a point further from the center of rotation (larger r) will have a greater linear velocity (v).
Practical Examples
Consider a rotating carousel:
- Everyone on the carousel completes one full rotation in the same amount of time. This means everyone has the same angular velocity, regardless of whether they are sitting near the center or on the outer edge.
- However, a person on the outer edge travels a larger distance (circumference) in that same amount of time compared to someone near the center. Therefore, the person on the outer edge has a greater linear velocity.
This illustrates clearly why angular velocity remains constant across different radii of a rigid rotating body, while linear velocity changes proportionally with the radius.
Summary Table
| Property | Measures | Dependence on Radius (r) | Formula Relationship (for constant ω) |
|---|---|---|---|
| Angular Velocity (ω) | How fast the angle changes | No dependence | ω = θ/t (constant for all r) |
| Linear Velocity (v) | How fast arc length changes | Direct dependence | v = rω (v increases as r increases) |
In conclusion, based on the definition provided and the nature of rotational motion, the angular velocity remains constant across different radii for a rigid rotating body.
