Linear and angular velocity
Linear and Angular Velocity
Linear and Angular Velocity is a topic in Trigonometric Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.4(D), which requires students to represent angles in applications of linear and angular velocity.
Angular velocity \(\omega=\dfrac{\theta}{t}\) measures how fast an angle turns, and linear velocity \(v=r\omega\) measures how fast a point on the rim travels.
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.