Calculator
Example Data Table
| # | Scenario | m (kg) | r (m) | v (m/s) | Position | Computed T (N) |
|---|---|---|---|---|---|---|
| 1 | Horizontal | 2 | 1.5 | 8 | — | 85.3333 |
| 2 | Vertical | 1 | 0.75 | 6 | Bottom | 57.8067 |
| 3 | Vertical | 1 | 0.75 | 6 | Top | 38.1934 |
| 4 | Horizontal | 0.25 | 0.4 | 12 | — | 90 |
| 5 | Vertical | 3 | 2 | 10 | Side | 150 |
Formula Used
- Fc = m · v² / r (centripetal force)
- aₙ = v² / r (centripetal acceleration)
- v = ω · r, ω = 2πf = 2π/P, RPM = 60f
- Horizontal circle: T = Fc
- Vertical circle: T = Fc + mg (bottom), T = Fc (side), T = Fc − mg (top)
- Conical pendulum: T = mg / cosθ, v = √(r g tanθ), tanθ = v²/(r g)
How to Use This Calculator
- Select what you want to solve for in the Solve for menu.
- Choose a Scenario. If using vertical motion, select Top, Side, or Bottom.
- Enter known inputs with their units. Provide a motion rate using speed, ω, period, frequency, or RPM.
- Press Calculate. Results will appear above the form under the header.
- Use Download CSV or Download PDF to save the results.
Centripetal Force and Tension Guide
1) Circular motion in one equation
Centripetal force is the inward net force needed to keep an object moving in a circle. For steady motion, Fc = m·v²/r and aₙ = v²/r. For m=2 kg, v=8 m/s, r=1.5 m, Fc≈85.33 N and aₙ≈42.67 m/s².
2) When tension equals the inward force
In a horizontal circle, tension provides the inward force, so T = Fc. Doubling speed increases force by four times because v is squared. Increasing radius reduces force linearly. Rearranging the same equation lets you solve for v, r, or m. You can also compute v from ω, period, frequency, or RPM using v = ωr.
3) Typical ranges you can test
For lab setups, masses from 0.1–3 kg and radii from 0.2–2 m are common. Speeds of 2–15 m/s cover gentle swings to fast rotations. At m=0.25 kg, r=0.40 m, v=12 m/s, Fc is 90 N. Record your values carefully.
4) Vertical circles add weight effects
In a vertical circle, Fc is still required, but tension changes with position. Bottom: T = Fc + mg. Side: T = Fc. Top: T = Fc − mg. If Fc is smaller than mg at the top, the string can go slack. Top condition: v ≥ √(g r) keeps tension non‑negative.
5) Conical pendulum data and angle impact
For a conical pendulum, θ is measured from vertical. Tension is T = mg/cosθ, and speed is v = √(r g tanθ). For r=1 m, θ=30° gives v≈2.38 m/s, while θ=60° gives v≈4.12 m/s.
6) Unit handling and sensitivity checks
This calculator converts inputs into consistent base units before solving. As a quick check, 1 kg at 10 m/s and 1 m radius gives Fc≈100 N. Large mismatches usually come from speed units, like km/h entered as m/s.
7) Practical uses and safer decisions
Use these outputs to size cords, estimate belt loads, and evaluate rotating rigs. Include a safety factor because real systems add drag, vibration, and peak loads. If top tension approaches zero, reduce speed or increase radius. Useful for ride design.
FAQs
1) Is centripetal force the same as tension?
No. Centripetal force is the required inward net force. Tension can provide all or part of that net force, but in vertical motion weight also contributes, so tension may be higher or lower than Fc.
2) Why does the calculator show slack at the top?
At the top of a vertical circle, tension is Fc − mg. If v²/r is less than g, the computed tension becomes negative, meaning the string cannot push and would go slack.
3) Which motion input should I use: speed, ω, period, frequency, or RPM?
Use whichever you measured directly. The tool converts between them using v = ωr, ω = 2πf, and f = 1/P. Enter only one to avoid conflicting values.
4) Can I solve for mass from tension?
Yes. In horizontal motion, m = Tr/v². In vertical motion, the formula changes because mg adds or subtracts depending on position. The calculator applies the correct relationship automatically.
5) How accurate are the results?
They are as accurate as your inputs. The formulas assume steady circular motion and a constant radius. Real systems can deviate due to stretch, air resistance, and changing speed, so add a safety margin for design work.
6) What angle does θ represent for the conical pendulum?
θ is measured from the vertical line, not from the horizontal plane. A small θ means a near-vertical string and lower speed. As θ increases toward 90°, tension rises sharply as cosθ decreases.