Linear method: plots tangential speed v(t) using computed aₜ and v₁.
| Timestamp | Method | Solve for | r (m) | α (rad/s²) | aₜ (m/s²) | v₁ (m/s) | v₂ (m/s) | t (s) |
|---|---|---|---|---|---|---|---|---|
| No saved calculations yet. | ||||||||
| Scenario | Inputs | Output |
|---|---|---|
| Rotational, compute aₜ | r = 0.35 m, α = 4.2 rad/s² | aₜ = 1.47 m/s² |
| Rotational, compute α | r = 0.50 m, aₜ = 2.25 m/s² | α = 4.50 rad/s² |
| Linear, compute aₜ | v₁ = 6 m/s, v₂ = 14 m/s, t = 3.5 s | aₜ = 2.2857 m/s² |
| Linear, compute v₂ | v₁ = 10 m/s, aₜ = 1.6 m/s², t = 4 s | v₂ = 16.4 m/s |
- Rotational form: aₜ = r · α
- Linear change form: aₜ = (v₂ − v₁) / t
- Rearrangements: α = aₜ / r, r = aₜ / α, v₂ = v₁ + aₜ·t, v₁ = v₂ − aₜ·t, t = (v₂ − v₁) / aₜ
- Choose the method that matches your available data.
- Select what you want to solve for.
- Enter the known values and pick their units.
- Click Calculate to view results and checks.
- Use the saved table to download CSV or PDF.
Tangential Acceleration Guide
What tangential acceleration means
Tangential acceleration, aₜ, measures how fast tangential speed changes along a circular path. It acts along the direction of motion, while centripetal acceleration points inward. In a wheel, belt pulley, or rotating arm, aₜ is the part that “speeds up” or “slows down” the rim. If rotation is steady, aₜ becomes zero even though centripetal acceleration remains.
Use the rotational relationship
When you know radius r and angular acceleration α, the calculator uses aₜ = r·α. For example, r = 0.35 m and α = 4.2 rad/s² gives aₜ = 1.47 m/s². Doubling radius doubles aₜ for the same α, so larger rotors create stronger tangential effects at the rim. If you enter α in deg/s² or rev/s², it is converted to rad/s² before multiplying by r.
Use the linear change relationship
If you have speeds and time, the calculator uses aₜ = (v₂ − v₁)/t. With v₁ = 6 m/s, v₂ = 14 m/s, and t = 3.5 s, aₜ = 2.2857 m/s². This method is useful for motion data from sensors, track tests, or video analysis. For best accuracy, use averaged speeds over the same interval and keep units consistent across v₁ and v₂.
Solve for any missing variable
Advanced mode lets you solve for α or r in the rotational form, and for v₁, v₂, or t in the linear form. Rearrangements include α = aₜ/r and v₂ = v₁ + aₜ·t. If a value computes negative, it indicates direction opposite your positive sign convention. In practice, treat magnitude as the size of the change, then apply your chosen clockwise or counter‑clockwise sign separately.
Unit conversion and practical scales
The calculator converts common units to SI internally. Radius can be entered in m, cm, mm, km, ft, in, or yd. Speeds accept m/s, km/h, mph, and ft/s (1 mph ≈ 0.44704 m/s). Acceleration can be shown in m/s², ft/s², in/s², or g (1 g = 9.80665 m/s²). Use g when comparing to comfort or vibration limits, and use m/s² for formulas.
Data checks and interpretation tips
Keep radius and time positive. Avoid α = 0 when solving for r, and avoid aₜ = 0 when solving for time. Compare results to expected ranges: aₜ of 0.5–5 m/s² is typical for many lab turntables, while vehicles and machinery can exceed 10 m/s² at large radii or fast ramps. If your plot looks flat, check whether your α or Δv is near zero, or whether the selected units are masking small changes.
Q1. Can tangential acceleration be negative?
Yes. A negative value means tangential speed decreases in your chosen positive direction. It can also indicate rotation in the opposite direction, depending on your sign convention.
Q2. What is the difference between tangential and centripetal acceleration?
Tangential acceleration changes speed along the path. Centripetal acceleration changes direction toward the center. You can have one without the other, or both at the same time.
Q3. Which method should I choose?
Use the rotational method when you know radius and angular acceleration. Use the linear method when you measured speeds over a time interval. Both give the same aₜ when inputs describe the same motion.
Q4. Why does the Plotly chart show a straight line?
In the rotational view, aₜ is proportional to r for a fixed α, so the relationship is linear. In the linear view, v(t) is linear for constant aₜ.
Q5. What units should I use for best accuracy?
Enter values in any supported units, but keep your measurements consistent. SI units reduce rounding and make comparisons easier. If you use mph or feet, ensure your source data is recorded in those units.
Q6. Does this calculator include friction or torque?
No. It computes kinematics only. To include dynamics, you would also need mass moment of inertia, torque, and losses. You can still use aₜ results as inputs to a dynamics model.