1. What is Angular Motion?
Angular motion refers to the rotation of a body around an axis. Unlike linear motion, which involves movement along a straight path, angular motion involves turning, spinning, or rotating. Everyday motions such as a ceiling fan spinning, a wheel turning, or a merry-go-round rotating are all examples of angular motion.
To describe this rotation mathematically, we use two key quantities: angular velocity and angular acceleration.
1.1. Why Do We Need These Quantities?
Just as linear motion uses velocity and acceleration to describe how fast and how quickly motion changes, rotational motion requires corresponding quantities that describe the speed of rotation and its rate of change.
1.1.1. Real-Life Intuition
- A fan's speed setting controls its angular velocity.
- A spinning top slowing down shows angular deceleration.
- An athlete spinning in a dance or skating routine changes angular velocity by adjusting their body posture.
2. Angular Velocity
Angular velocity measures how fast a body is rotating. It tells us how much the angular position changes per unit time. The direction of angular velocity is along the axis of rotation, following the right-hand rule.
The average angular velocity is defined as:
\( \omega = \dfrac{\Delta \theta}{\Delta t} \)
Where \( \Delta \theta \) is the change in angular displacement and \( \Delta t \) is the time interval.
2.1. Instantaneous Angular Velocity
The instantaneous angular velocity is defined as the rate of change of angular displacement at a particular moment:
\( \omega = \dfrac{d\theta}{dt} \)
2.2. Units and Direction
The SI unit of angular velocity is radian per second (rad/s). Its direction is determined by the right-hand rule: curling your fingers in the direction of rotation, your thumb points along the direction of \( \omega \).
3. Angular Acceleration
Angular acceleration measures how quickly the angular velocity changes with time. A body speeds up or slows down its rotation based on the torque acting on it.
Angular acceleration is defined as:
\( \alpha = \dfrac{\Delta \omega}{\Delta t} \)
3.1. Instantaneous Angular Acceleration
The instantaneous angular acceleration is given by:
\( \alpha = \dfrac{d\omega}{dt} \)
This tells us the exact rate of change of angular velocity at a moment.
3.2. Relation with Torque
Angular acceleration is related to torque through the rotational form of Newton's second law:
\( \tau = I \alpha \)
This means that for a given torque, a body with a smaller moment of inertia will have a larger angular acceleration.
4. Kinematic Equations for Rotational Motion
Just like linear motion has kinematic equations, rotational motion has similar equations that relate angular displacement, angular velocity, angular acceleration, and time (for constant angular acceleration).
4.1. Rotational Kinematic Equations
- \( \omega = \omega_0 + \alpha t \)
- \( \theta = \omega_0 t + \dfrac{1}{2} \alpha t^2 \)
- \( \omega^2 = \omega_0^2 + 2 \alpha \theta \)
5. Connection Between Linear and Angular Quantities
Rotational motion and linear motion are closely related. When a point on a rotating body moves, it follows a circular path. The following relations connect linear and angular variables:
5.1. Key Relations
- Linear velocity: \( v = r \omega \)
- Linear acceleration: \( a = r \alpha \)
- Centripetal acceleration: \( a_c = r \omega^2 \)
These relations show that the behaviour of rotating bodies can be understood using both linear and angular concepts.
6. Why Angular Velocity and Acceleration Matter
Angular velocity and angular acceleration help describe how bodies rotate, speed up, or slow down. These quantities are essential in understanding the behaviour of wheels, gears, turbines, fans, planets, and countless rotating systems.
They also connect directly with torque, moment of inertia, and energy — topics that help build a complete understanding of rotational mechanics.