What is Angular Velocity?
In physics, while linear velocity describes how fast an object moves along a straight line, angular velocity describes the rate at which an object rotates or revolves around an axis. It measures how quickly the angular position of a body changes with respect to time.
The Definition
Angular velocity (denoted by the Greek letter $\omega$, omega) is defined as the rate of change of the angular displacement $\theta$. It represents how many radians an object rotates through in a given time interval.
Mathematical Formula
For an object moving in a circle, the average angular velocity is given by:
$$\omega = \frac{\Delta \theta}{\Delta t}$$
Where:
- $\omega$ = Angular velocity (measured in radians per second, rad/s).
- $\Delta \theta$ = Change in the angular position (in radians).
- $\Delta t$ = Change in time (in seconds).
If we consider an instantaneous change, we use calculus:
$$\omega = \frac{d\theta}{dt}$$
Intuition and Real-World Examples
Think of a spinning record or a rotating wheel. Every point on the wheel completes a full rotation ($2\pi$ radians) in the same amount of time. If it spins faster, the time $\Delta t$ decreases, making $\omega$ larger.
- Units: The SI unit is radians per second ($rad/s$). Sometimes it is measured in revolutions per minute ($rpm$).
- Vector Nature: Angular velocity is a vector. Its direction is determined by the Right-Hand Rule: if you curl the fingers of your right hand in the direction of the rotation, your thumb points in the direction of the angular velocity vector.
Relationship to Linear Velocity
There is a direct relationship between angular velocity ($\omega$) and linear velocity ($v$) for a point at a distance $r$ from the axis of rotation:
$$v = r \cdot \omega$$
This shows that even if two points on a rotating disc have the same angular velocity, the point further from the center (larger $r$) has a higher linear velocity.