Library / Mechanical / MEC-14
Mechanical
Rolling without slipping
Speed of a wheel centre from its spin rate and radius.
- Ref
- MEC-14
- Rev
- v1
- Field
- Mechanical
- Result
- vG [m/s]
A wheel of radius R rolling on a fixed straight surface. The contact point has zero instantaneous velocity (but can have non-zero acceleration). ω in rad/s.
A wheel that rolls without slipping moves forward by one circumference per turn.
Calculator
m/s
m
rad/s
- Speed of the top of the wheel6 m/s
- Spin rate95.49 rpm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Centre speed (m/s) current value 3 m/s
ωSpin rateKey driver
0.0360
RWheel radius
0.220
- The top of the wheel moves at twice the centre speed; the contact point is momentarily at rest.
- If the wheel skids the relation fails and vG is not Rω.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Spin rate | 0.1 to 200 rad/s | 0.03 | 60 | 59.97 | +10% | +10 % |
| R Wheel radius | 0.02 to 2 m | 0.2 | 20 | 19.8 | +10% | +10 % |
Sweep of Spin rate
| Spin rate (rad/s) | Centre speed (m/s) |
|---|---|
| 0.1 | 0.03 |
| 0.2586 | 0.07758 |
| 0.6687 | 0.2006 |
| 1.729 | 0.5188 |
| 4.472 | 1.342 |
| 11.56 | 3.469 |
| 29.91 | 8.972 |
| 77.34 | 23.2 |
| 200 | 60 |
Sweep one variable
Centre speed (m/s)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
vG
Centre speed
What the equation gives you. Everything else feeds it.
R
Wheel radius
Radius of the rolling wheel. Directly proportional.
ω
Spin rate
Angular speed of the wheel. Directly proportional.
Worked example with the values above
- R = 0.3 m; ω = 10 rad/s.
- vG = Rω = 0.3 × 10.
- vG = 3 m/s.
Assumptions and limits
- No slip at the contact.
- Rigid wheel on a fixed straight surface.
Common mistakes
- Using diameter in place of radius.
- Forgetting the radian conversion for rpm.
Statistics and uncertainty
Tyre radius changes with load and wear, which makes speed from wheel rotation uncertain by 1–2%.
Where it is used
- Odometry and vehicle speed sensing.
- Wheeled and tracked vehicle dynamics.
Maths you need
- Angular speed: rad/s
- Circle: circumference 2πR