Short Answer:
Angular velocity is defined as the rate of change of angular displacement with respect to time. It tells how fast an object rotates or revolves around a fixed axis. In other words, it measures how quickly the angle through which a body rotates changes with time.
It is generally represented by the symbol ω (omega) and its standard unit is radian per second (rad/s). Angular velocity is a vector quantity, having both magnitude and direction. The direction of angular velocity is determined by the right-hand rule — curling the fingers in the direction of rotation, the thumb shows the direction of angular velocity.
Detailed Explanation :
Angular Velocity
In mechanics, when a rigid body rotates about a fixed axis, all the points on that body move in circular paths centered on the axis of rotation. The angular velocity of the rotating body describes how fast the body is rotating. It tells us the rate at which the angular position of a body changes with respect to time.
If a rotating object sweeps an angular displacement (in radians) in a certain time , then the average angular velocity is given by:
For very small intervals of time, the instantaneous angular velocity is the limiting value of this ratio as approaches zero:
Angular velocity represents the speed of rotation. The greater the angular velocity, the faster the body rotates.
Units and Dimensions
- SI Unit: radian per second (rad/s)
- Other Units: degree per second (°/s), revolution per minute (rpm)
- Dimensional Formula:
The radian per second is most commonly used in engineering and physics because radians are a natural measure of angular displacement.
Nature of Angular Velocity
Angular velocity is a vector quantity. It has:
- Magnitude: which shows how fast the object is rotating.
- Direction: which shows the axis and sense of rotation.
The direction is determined using the right-hand rule. If the rotation is counterclockwise when viewed from above, the angular velocity vector points upward along the axis. If the rotation is clockwise, it points downward.
Relation Between Linear and Angular Velocity
A point on a rotating body also has linear velocity due to its circular motion. The magnitude of this linear velocity depends on both angular velocity and the distance from the axis of rotation. The relation between linear velocity and angular velocity is:
where,
= linear velocity (m/s)
= radius of the circular path (m)
= angular velocity (rad/s)
This equation shows that points farther from the axis move faster in terms of linear speed, even though their angular velocity is the same.
Example of Angular Velocity
- Rotating Fan:
The blades of a fan rotate with a certain angular velocity around the center shaft. Each blade point moves in a circle, and the angle it covers per second defines the angular velocity. - Earth’s Rotation:
The Earth rotates about its own axis once in 24 hours. Its angular velocity is:
This shows how slowly the Earth spins when measured in radians per second.
- Wheels of a Vehicle:
When a car moves on the road, its wheels rotate with angular velocity proportional to the car’s speed and inversely proportional to the wheel’s radius.
Types of Angular Velocity
- Uniform Angular Velocity:
When a body rotates through equal angles in equal intervals of time, the angular velocity remains constant. Example: A ceiling fan running at a constant speed. - Variable Angular Velocity:
When a body rotates through unequal angles in equal intervals of time, its angular velocity changes with time. Example: A fan gradually increasing its speed after being switched on.
Graphical Representation
If we plot a graph between angular displacement (θ) and time (t):
- For uniform angular velocity, the graph is a straight line with a constant slope.
- For variable angular velocity, the graph is curved, and the slope at any point gives the instantaneous angular velocity.
Similarly, if we plot angular velocity (ω) against time (t), the slope of this graph gives angular acceleration (α):
Relation Between Angular Quantities
There are direct analogies between linear motion and rotational motion. These relations help engineers to analyze rotating systems easily:
| Linear Motion | Rotational Motion |
| Displacement (s) | Angular displacement (θ) |
| Velocity (v) | Angular velocity (ω) |
| Acceleration (a) | Angular acceleration (α) |
| Force (F) | Torque (τ) |
| Mass (m) | Moment of inertia (I) |
This comparison helps in understanding rotational systems like gears, turbines, and wheels using similar principles to those used in linear motion.
Applications of Angular Velocity
Angular velocity plays a very important role in mechanical engineering. It is widely used in:
- Rotating machinery such as turbines, engines, gears, and motors.
- Design of vehicles to determine wheel and crank speeds.
- Robotics to control the speed of rotating joints.
- Aerospace engineering to study the spin and rotation of satellites and aircraft.
- Kinematics of mechanisms like cams, levers, and gear trains.
Engineers often convert between angular and linear velocities to analyze machine performance and ensure balanced, safe operation.
Conclusion
Angular velocity is the rate at which an object rotates about a fixed axis. It defines how fast an angular displacement changes with time and gives both the speed and direction of rotation. Expressed in radians per second, it forms the foundation for analyzing rotational motion. Angular velocity is used in various engineering applications involving wheels, gears, turbines, and rotating machinery. Understanding it helps in designing efficient systems that rely on rotational motion and control.