Library / Mechanical / MEC-13
Mechanical
Transverse acceleration in polar coordinates
Tangential component of acceleration for a point in planar polar coordinates.
- Ref
- MEC-13
- Rev
- v1
- Field
- Mechanical
- Result
- aθ [m/s²]
Tangential direction of increasing angle. The term 2ṙθ̇ is the Coriolis contribution: it appears when the point moves radially on a rotating line.
A point moving outward along a spinning arm is pushed sideways by the rotation; the angular acceleration adds to this.
Calculator
- Angular term r θ̈2 m/s²
- Coriolis term 2 ṙ θ̇3 m/s²
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.
- A radial speed of only 0.5 m/s on an arm turning at 3 rad/s adds 3 m/s² of sideways acceleration.
- Both terms vanish for a point at rest in a non-rotating frame.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ṙ Radial speed | 0.01 to 10 m/s | 2.06 | 62 | 59.94 | +10% | +6 % |
| θ̈ Angular acceleration | 0.01 to 20 rad/s² | 3.02 | 43 | 39.98 | +10% | +4 % |
| r Radius | 0.1 to 20 m | 3.1 | 23 | 19.9 | +10% | +4 % |
| θ̇ Angular rate | 0.1 to 20 rad/s | 2.1 | 22 | 19.9 | +10% | +6 % |
Sweep of Angular acceleration
| Angular acceleration (rad/s²) | Transverse acceleration (m/s²) |
|---|---|
| 0.01 | 3.02 |
| 2.509 | 8.017 |
| 5.007 | 13.01 |
| 7.506 | 18.01 |
| 10 | 23.01 |
| 12.5 | 28.01 |
| 15 | 33 |
| 17.5 | 38 |
| 20 | 43 |
Sweep one variable
Variables and how each one contributes
Transverse acceleration
What the equation gives you. Everything else feeds it.
Radius
Distance from the origin. Scales the angular-acceleration term.
Radial speed
Rate of change of the radius. Contributes the Coriolis term with θ̇.
Angular rate
Angular velocity. Scales the Coriolis term.
Angular acceleration
Rate of change of angular velocity. Scales with the radius.
Worked example with the values above
- Angular term: rθ̈ = 2 × 1 = 2 m/s².
- Coriolis term: 2ṙθ̇ = 2 × 0.5 × 3 = 3 m/s².
- aθ = 5 m/s².
Assumptions and limits
- Fixed origin.
- Consistent sign conventions for ṙ and θ̇.
Common mistakes
- Dropping the factor 2 on the Coriolis term.
- Mixing the sign of ṙ and θ̇.
Statistics and uncertainty
The Coriolis term multiplies two measured rates, so their relative errors add.
Where it is used
- Rotating mechanisms and slider cranks.
- Missiles and radars tracked in polar coordinates.
Maths you need
- Polar coordinates: r and θ
- Product rule: differentiating rθ̇