Angular Velocity (College Board AP® Physics 1: Algebra-Based)

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Katie M

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Physics

Angular velocity

Angular speed is defined as:

The change in angular position per unit time

In other words, it describes the rate of change of a system's angular position

Diagram showing a circle with points A, B, and C. Line AB rotates to AC by angle Δθ. Time taken is Δt. ω = Δθ/Δt. Descriptions in boxes. — **The angular speed ω is the rate at which the line AB rotates to line AC by angular displacement Δθ**

Average angular velocity

The average angular velocity of a rigid rotating system is defined as:

The average rate at which angular position changes with respect to time

Average angular velocity considers the initial and final states of a system over an interval of time
- In other words, the change in angular position over the time interval for which the system rotated
This can be expressed as an equation:

$ω_{a v g} = \frac{∆ θ}{∆ t}$

Where:
- $ω_{a v g}$ = average angular velocity, in $rad / s$
- $∆ θ$ = angular displacement, in $rad$
- $∆ t$ = change in time, in $s$
This can also be expressed as:

$ω_{a v g} = \frac{θ - θ_{0}}{∆ t}$

Where:
- $θ$ = final angular position, in $rad$
- $θ_{0}$ = initial angular position, in $rad$

Direction of angular velocity

Like linear velocity, angular velocity is a vector quantity with both magnitude and direction
- Angular velocity acts in the same direction as the angular displacement
For example, consider a disc rotating about an axis of rotation at its center, where counterclockwise is defined as the positive direction
- If the disc rotates counterclockwise, its angular velocity is positive
- If the disc rotates clockwise, its angular velocity is negative

Two diagrams show counterclockwise (left, positive direction) and clockwise (right, negative direction) rotations with labeled angles, reference lines, and rotational arrows. — **If counterclockwise is defined as the positive direction, then when the disc rotates counterclockwise, its angular velocity is positive, and negative when it rotates clockwise**

Units of angular velocity

Angular velocity is measured in radians per second or $rad / s$
- Since radians can be omitted, it can also be written as $1 / s$
It is also sometimes expressed in units of revolutions per minute, or rpm
The rotation angle for one complete revolution is equal to

1 revolution = $360 °$ = $2 π radians$

Therefore, to convert from rpm to rad/s, multiply by 2π radians per revolution and divide by 60 seconds per minute:

$ω (rad / s) = ω (rpm) \times \frac{2 π rad}{60 s}$

Worked Example

What is the angular velocity, in rad/s, of a flywheel that spins at a rate of 1200 rpm?

Answer:

Step 1: Analyze the scenario

A rate of 1200 rpm means the flywheel completes 1200 revolutions per minute

Step 2: Convert from rpm to rad/s

There are 2π radians in one complete revolution and 60 seconds in one minute, so its angular velocity is:

$ω = 1200 \times \frac{2 π}{60} = 40 π = 126 rad / s$

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