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

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

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:

${\omega }_{avg}=\frac{∆\theta }{∆t}$

• Where:

  • ${\omega }_{avg}$ = average angular velocity, in $\mathrm{rad}/\mathrm{s}$

  • $∆\theta$ = angular displacement, in $\mathrm{rad}$

  • $∆t$ = change in time, in $\mathrm{s}$

• This can also be expressed as:

${\omega }_{avg}=\frac{\theta -{\theta }_0}{∆t}$

• Where:

  • $\theta$ = final angular position, in $\mathrm{rad}$

  • ${\theta }_0$ = initial angular position, in $\mathrm{rad}$

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

Units of angular velocity

• Angular velocity is measured in radians per second or $\mathbf{rad}\mathbf{/}\mathbf{s}$

  • Since radians can be omitted, it can also be written as $\mathbf{1}\mathbf{/}\mathbf{s}$

• 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 = $360°$ = $2\pi \mathrm{radians}$

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

$\omega \left(\mathrm{rad}/\mathrm{s}\right)=\omega \left(\mathrm{rpm}\right)\times \frac{2\mathrm{\pi }\mathrm{rad}}{60\mathrm{s}}$