Linear and angular velocity
Theory
When something rotates, two speeds describe it:
- Angular velocity \(\omega\) — how fast the angle changes: \(\omega=\dfrac{\theta}{t}\), in radians per unit time.
- Linear velocity \(v\) — how fast a point on the rim travels along its circular path: distance per unit time.
They are connected through the radius:
The rotation relationships:
How to solve rotation problems
- Convert revolutions to radians (\(\times 2\pi\)) and fix the time units.
- Find \(\omega\) in rad per unit time.
- Apply \(v=r\omega\) for the linear speed, or \(s=r\theta\) for distance.
- Rearrange if you need \(r\), \(\omega\), or \(t\) instead.
Each revolution is \(2\pi\) rad; convert minutes to seconds.
| \(\omega\) | \(=\) | \(30\cdot\dfrac{2\pi\ \text{rad}}{60\ \text{s}}\) |
| \(=\) | \(\pi\ \text{rad/s}\) |
Use \(v=r\omega\).
| \(v\) | \(=\) | \(0.5\cdot\pi\) |
| \(=\) | \(\dfrac{\pi}{2}\ \text{m/s}\approx 1.57\ \text{m/s}\) |
Solve \(v=r\omega\) for \(r\).
| \(r\) | \(=\) | \(\dfrac{v}{\omega}=\dfrac{20}{8}\) |
| \(=\) | \(2.5\ \text{ft}\) |
Arc length \(s=r\theta\).
| \(s\) | \(=\) | \(3\cdot\dfrac{4\pi}{3}\) |
| \(=\) | \(4\pi\ \text{ft}\) |
Common pitfalls
Frequently asked questions
What is the difference between linear and angular velocity?
Angular velocity is how fast the angle turns (radians per time); linear velocity is how fast a rim point moves along its path (distance per time). They relate by \(v=r\omega\).
How do you convert rpm to radians per second?
Multiply revolutions by \(2\pi\) radians and divide by 60 seconds: \(\text{rpm}\times\dfrac{2\pi}{60}\).
What is the formula linking the two velocities?
\(v=r\omega\): linear velocity equals radius times angular velocity, with \(\omega\) in radians per unit time.
Why must angles be in radians here?
Because \(v=r\omega\) and \(s=r\theta\) come from the arc-length relationship, which only holds when the angle is in radians.