Calculator Inputs
Example Data Table
| Scenario | m (kg) | r (m) | v (m/s) | Angle | Estimated Tension (N) |
|---|---|---|---|---|---|
| Vertical, bottom | 2.0 | 0.8 | 6.0 | θ = 0° | ≈ 109.6 |
| Vertical, side | 2.0 | 0.8 | 6.0 | θ = 90° | ≈ 90.0 |
| Vertical, top | 2.0 | 0.8 | 6.0 | θ = 180° | ≈ 70.4 |
| Horizontal circle | 1.5 | 1.0 | 5.0 | — | ≈ 37.5 |
| Conical pendulum | 1.0 | — | Computed | α = 30° | ≈ 11.32 |
Values are rounded examples. Your exact result depends on your inputs and gravity.
Formula Used
Core idea: centripetal acceleration is ac = v² / r, and centripetal force is Fc = m ac.
Horizontal circle: T = m(v²/r)
Conical pendulum (α from vertical): T = mg / cos(α)
Conical required speed: v = √(r g tan(α)), where r = L sin(α)
Angles use degrees in the form, but are converted internally to radians for trigonometry.
How to Use This Calculator
- Select a mode that matches your setup.
- Enter mass and choose the correct mass unit.
- For vertical or horizontal motion, enter the circle radius.
- Pick how you want to provide motion data (speed, RPM, period).
- For vertical motion, set θ from the bottom point.
- For conical motion, enter string length and α from vertical.
- Press Calculate to view tension and related quantities.
- Use the CSV or PDF buttons to save results.
Circular Motion Tension Guide
This calculator estimates the tension needed to keep a mass on a circular path. It reports tension in newtons and pounds-force, plus speed, angular speed, period, frequency, and g-load. It also highlights when slack is likely at the top.
1) Core relationship: v² over r
Centripetal acceleration is ac = v²/r. Because v is squared, small speed changes matter. Example: v = 8 m/s and r = 2 m gives ac = 32 m/s², about 3.26 g (using g = 9.80665 m/s²).
2) Force level depends on mass
Centripetal force is Fc = m ac. With the same 32 m/s² acceleration, a 1 kg mass needs 32 N inward, while a 5 kg mass needs 160 N. This is why heavier loads demand stronger strings at the same motion.
3) Vertical circle: gravity changes tension
For a vertical loop the calculator uses T = m(v²/r) + mg cos(θ), where θ is measured from the bottom. At θ = 0° the mg term adds; at θ = 180° it subtracts. That difference can be 2mg between bottom and top.
4) Top slack check and minimum speed
Near the top, a common “taut string” rule is v ≥ √(gr). If r = 0.80 m, √(gr) ≈ 2.80 m/s. Below that, tension can drop to zero, meaning the string cannot pull inward and the motion is no longer circular.
5) Horizontal circle: simplest tension model
In horizontal mode, tension supplies the whole inward force: T = m(v²/r). Example: m = 1.5 kg, v = 5 m/s, r = 1 m → T = 37.5 N. Use this for flat turns, whirling masses, or quick lab estimates.
6) Conical pendulum: angle controls everything
For a steady cone with angle α from vertical, the calculator uses T = mg/cos(α). At α = 30°, cos(α) ≈ 0.866 so tension ≈ 1.155 mg. The circle radius is r = L sin(α), and the speed follows v = √(r g tan(α)).
7) Inputs, conversions, and reporting
You can enter speed as m/s, km/h, mph, ω, RPM, period, or frequency. For reference, 60 km/h ≈ 16.67 m/s and 300 RPM ≈ 31.42 rad/s. Export CSV or PDF to archive your run and share calculations cleanly.
FAQs
1) What does θ mean in vertical mode?
θ is measured from the bottom of the loop. Use 0° at the bottom, 90° at the side, and 180° at the top. The cos(θ) term models how gravity adds or subtracts from tension.
2) Which speed input should I choose?
Use the value you actually know. If you have motor speed, pick RPM. If you timed one revolution, pick period. If you know linear speed, choose m/s, km/h, or mph. The calculator converts everything to v and ω.
3) Why can tension be lower at the top?
At the top of a vertical circle, gravity points toward the center, helping provide inward force. That reduces how much the string must pull. If speed is too low, tension can reach zero and the string goes slack.
4) What is g-load in the results?
g-load is ac/g, the centripetal acceleration compared to standard gravity. A value of 3 means the motion demands three times Earth’s gravitational acceleration as inward acceleration.
5) Can I use this for banking turns or roller coasters?
You can use it as a physics estimate when a single inward force dominates. Real rides include normal forces, track geometry, friction, and changing speed. For design decisions, add safety factors and more detailed modeling.
6) Why are my results different from my spring scale?
Measurements include string mass, air drag, hand motion, and non-constant speed. Also, a scale reads along its axis and may oscillate. Use steady conditions, average readings, and confirm your radius and speed inputs.