Trigonometry / Unit 1: Angles and Their Measure
1.5 Angular and Linear Speed
Notes
An object moving in a circle has angular speed $\omega$ (how fast the angle changes) and linear speed $v$ (how fast it travels along the arc).
- $\displaystyle \omega = \frac{\theta}{t}$, with $\theta$ in radians.
- $\displaystyle v = \frac{s}{t} = r\omega$.
- One revolution is $2\pi$ radians, so $n$ revolutions per minute is $2\pi n$ radians per minute.
- Example: a wheel of radius 0.3 m turning at $10$ rad/s has $v = 0.3 \cdot 10 = 3$ m/s.
Practice
A wheel makes 3 full revolutions in 2 seconds. What is its angular speed?
$3$ revolutions $= 6\pi$ rad, and $\displaystyle \frac{6\pi}{2} = 3\pi$ rad/s.
A point on a wheel of radius 0.5 m turns at $8$ rad/s. What is its linear speed?
$v = r\omega = 0.5 \cdot 8 = 4$ m/s.
A record spins at 45 revolutions per minute. What is its angular speed in radians per minute?
Each revolution is $2\pi$ rad, so $45 \cdot 2\pi = 90\pi$.
Ready for more? Take the Unit 1 test.