4.7 Angular Velocity

Introduction

Angular velocity is the rotational analog of linear velocity. While linear velocity describes the rate of change of an object's position, angular velocity describes the rate of change of its angular displacement. It quantifies how fast an object is spinning or revolving around an axis and in what direction.

Definition and Formula

Angular velocity ($\omega$) is defined as the rate at which angular displacement ($\theta$) changes with respect to time ($t$).

Average Angular Velocity (${\omega }_{avg}$): The total angular displacement divided by the total time interval. ${\omega }_{avg}=\frac{\Delta \theta }{\Delta t}=\frac{{\theta }_f-{\theta }_i}{t_f-t_i}$

Instantaneous Angular Velocity (${\omega }_{inst}$): The angular velocity at a specific moment in time. It is the limit of the average angular velocity as the time interval approaches zero. ${\omega }_{inst}={\lim }_{\Delta t\to 0}\frac{\Delta \theta }{\Delta t}$

In most introductory physics problems, if the rotation is steady, the average and instantaneous angular velocities are the same.

Vector Nature and Direction

Angular velocity is a vector quantity.

• Magnitude: The magnitude of the angular velocity is the object's angular speed.
• Direction: The direction of the angular velocity vector is along the axis of rotation and is determined by the right-hand rule:

1. Curl the fingers of your right hand in the direction of the object's rotation.
2. Your thumb will point in the direction of the angular velocity vector ($\vec{\omega }$).

By convention, counter-clockwise rotation in the xy-plane is often considered positive (pointing in the +z direction), and clockwise rotation is negative (pointing in the -z direction).

Units of Angular Velocity

Several units are used to measure angular velocity, and it is important to know how to convert between them.

Key conversion: $1\text{ rev}=360^{\circ }=2\pi \text{ rad}$

Dimensions of Angular Velocity

Since radian is a dimensionless unit (ratio of length to length), the dimensions of angular velocity are: $[\omega ]=[T^{-1}]$

Relationship to Linear (Tangential) Velocity

Every point on a rotating object also has a linear velocity, called tangential velocity ($v$), which is directed tangent to its circular path. The magnitude of the tangential velocity is related to the angular velocity by: $v=r\omega$ Where:

• $v$ is the tangential velocity
• $r$ is the distance of the point from the axis of rotation (the radius)
• $\omega$ is the angular velocity in radians per second

This formula shows that for a given angular velocity, points farther from the center move with a greater linear speed. This relationship is a fundamental part of Scalar And Vector Quantities→ and circular motion.

Relationship to Frequency and Period

Frequency (f): The number of revolutions per second (measured in Hertz, Hz). $\omega =2\pi f$

Period (T): The time taken for one complete revolution (measured in seconds). $\omega =\frac{2\pi }{T}$

Possible Questions and Answers

Q: What is the difference between angular velocity and linear velocity?

A: Angular velocity describes how fast an object is rotating (change in angle over time), and it is the same for every point on a rigid rotating body. Linear velocity describes how fast a point is moving along its path (change in distance over time), and it depends on how far the point is from the axis of rotation ($v=r\omega$).

Q: Can an object have a constant angular velocity but still be accelerating?

A: Yes. Even if the angular velocity is constant (zero angular acceleration), every point on the object is constantly changing direction as it moves in a circle. This change in the direction of the linear velocity vector means there is a centripetal acceleration directed towards the center of rotation.