The Speed Wheel
Linear and Angular Speed
Step 1 of 6
Two cyclists ride side by side, pedaling at the same cadence. One has 26-inch wheels, the other has 20-inch wheels. Same pedaling rate. Who is moving faster across the ground?
Step 2 of 6
Watch the wheels roll. A marked dot on each rim traces its path. Both wheels rotate at the same rate, but look at how much ground each one covers per rotation.
Step 3 of 6
Same rotation rate, different ground speed. The rotation rate is called angular speed. The ground speed is called linear speed. What connects them?
The bigger wheel has a larger radius. Same angular speed, larger radius, more distance per rotation.
Which quantity converts angular speed into linear speed?
Step 4 of 6
Let us derive the formula. From Simulation 5, the arc length traced by a point on the rim is:
In one second, the point sweeps angle \(\theta\) along an arc of length \(s\). If we divide both sides by time \(t\), what do we get?
Step 5 of 6
Now reverse the question. Two meshed gears: a small gear drives a larger gear. Where their teeth meet, the linear speed at the contact point must be identical (the teeth are locked together). Which gear spins faster?
Step 6 of 6
Synthesis: \( v = r\omega \) has three quantities. Fix any one and the other two trade off. For each scenario below, identify which quantity stays constant.
Scenario A: Two points on the same spinning disk, one near the center, one near the edge.
Scenario B: Two meshed gears of different sizes.
Scenario C: Putting bigger tires on the same axle spinning at the same RPM.
Explore: Spin the Wheel
Dot on rim traced in red. Radius shown as green line.
Adjust the radius and RPM sliders to see how they affect linear speed. The wheel radius is 14 inches and it spins at 60 RPM, giving a linear speed of 87.96 inches per second at the rim.
Try This
Starter: Tire Speed
Tires are 28 inches in diameter, turning at 800 RPM. How fast is the car moving in miles per hour?
Hint: Convert RPM to rad/s, multiply by the radius in inches, then convert inches per second to mph. (Within 2 mph.)
Stretch: Vinyl Record
A vinyl record spins at 33⅓ RPM. Find the needle's linear speed at the outer edge (6 inches from center) and near the label (2 inches from center).
Within 0.5 in/s for each answer.
The groove holds the same amount of audio information per inch of groove. What does this tell you about sound quality across the record?
Challenge: Wind Turbine
A wind turbine blade is 170 feet long and turns at 15 RPM. How fast is the tip of the blade moving in mph?
Within 5 mph.
The answer is over 180 mph. Now explain: why are turbine blades speed-limited by tip speed, not by RPM?
- Distinguish between angular speed (rotation rate) and linear speed (ground speed).
- Derive and apply the relationship \( v = r\omega \).
- Convert between RPM and radians per second.
- Solve real-world problems involving wheels, gears, and rotating objects.
Quick Check
Two gears are meshed together. Gear A has a radius of 4 cm and Gear B has a radius of 12 cm. If Gear A spins at 300 RPM, how fast does Gear B spin?
Instructor Notes
Teaching Notes
This simulation bridges arc length (Simulation 5) to the speed formula. The derivation \( s = r\theta \to v = r\omega \) should feel like a natural consequence, not a new formula to memorize. Emphasize that dividing both sides of the arc length formula by time is all it takes.
The gear example in Step 5 reverses the reasoning: same linear speed, different radii, so different angular speeds. Students who grasp both directions understand the formula more deeply than those who only apply it forward.
Use the RPM-to-rad/s conversion explicitly. Many students treat RPM as a "speed" without converting, which gives answers off by a factor of \( 2\pi/60 \).
Common Student Errors
- Treating RPM as a speed rather than a rate of rotation. Students plug RPM directly into \( v = r\omega \) without converting to rad/s.
- Forgetting that \(\omega\) must be in radians per second for \( v = r\omega \) to yield the correct linear speed.
- Confusing diameter with radius. The tire problem gives diameter; students who forget to halve it get double the answer.
- Unit chain errors in the Apply problems, especially converting inches per second to miles per hour.
Discussion Questions
- If you put 30-inch tires on a truck calibrated for 26-inch tires, will the speedometer read high or low? By what percentage?
- Why do large wind turbines turn slowly while small desk fans spin fast, even though both move air?
- A car's transmission changes the gear ratio. How does this relate to \( v = r\omega \)?
Exam Connection
Typical exam problems: given RPM and radius, find linear speed (with unit conversions). Gear ratio problems where two meshed gears share linear speed. Multi-step problems combining arc length and speed. Expect the RPM-to-rad/s conversion to appear in nearly every problem.