# PHYS 2211 Module 10.1

## Rotational Variables 10.1 Rotational Variables

### Learning Objectives

By the end of this section, you will be able to:

• Describe the physical meaning of rotational variables as applied to fixed-axis rotation
• Explain how angular velocity is related to tangential speed
• Calculate the instantaneous angular velocity given the angular position function
• Find the angular velocity and angular acceleration in a rotating system
• Calculate the average angular acceleration when the angular velocity is changing
• Calculate the instantaneous angular acceleration given the angular velocity function

### Angular Position A particle follows a circular path. As it moves counterclockwise, it sweeps out a positive angle 𝜃 with respect to the x-axis and traces out an arc length s.

In the figure above, the red dot is a particle that moves in uniform circular motion. Its position vector from the origin of the circle to the particle sweeps out the angle 𝜃, which increases in the counterclockwise direction as the particle moves along its circular path. The angle 𝜃 is called the angular position of the particle.

The angle is related to the radius of the circle and the arc length by:

The angular displacement is the change in angular position: .

### Angular Velocity

The magnitude of the angular velocity, denoted by 𝜔 (omega), is the time rate of change of the angle 𝜃 as the particle moves in its circular path.

The instantaneous angular velocity is defined as the limit in which in the average angular velocity:

### Angular Acceleration

We need to define an angular acceleration for describing situations where 𝜔 changes. The faster the change in 𝜔, the greater the angular acceleration.

We define the instantaneous angular acceleration 𝛼 as the derivative of angular velocity with respect to time: Practice!

### Relating Linear and Angular Measurements

Earlier in this module we defined the angular position as

Now let’s rearrange this to solve for the arc length, s:

The arc length, s, is measured in units of meters and is a linear measurement, while 𝜃 is measured in units of rad (radians) and is an angular measurement. This expression relates the linear position to the angular position.

Similarly, the linear speed (v) of a particle is related to the angular speed by: and the linear acceleration (a) is related to the angular acceleration by: . Practice! Discuss!

Rebecca has gone to a conference in Albany, NY, leaving Brent at home in Dahlonega to look after Max the dog.

(a) Which one them has the greater angular speed, ω?

(b) Which of them has the greater tangential speed, v ?

The small wheel in the picture is rotating at 50 rpm. What is the angular speed of the large wheel if its radius is twice that of the small wheel.