Calculator Input
Enter consistent SI units. Choose a motion case, then calculate the required rope tension.
Example Data Table
These sample values use mass 2 kg, radius 1.5 m, speed 6 m/s, and gravity 9.81 m/s².
| Case | Angle from top | Tension formula | Result |
|---|---|---|---|
| Vertical top | 0° | T = mv²/r - mg | 28.38 N |
| Vertical side | 90° | T = mv²/r | 48.00 N |
| Vertical bottom | 180° | T = mv²/r + mg | 67.62 N |
| Vertical custom | 60° | T = mv²/r - mg cos(60°) | 38.19 N |
| Flat horizontal | Not used | T = mv²/r | 48.00 N |
Formula Used
Tension provides some or all of the inward centripetal force. The exact equation depends on the motion geometry.
T + mg cos(θ) = mv²/r
T = mv²/r - mg cos(θ)
Top: T = mv²/r - mg
Side: T = mv²/r
Bottom: T = mv²/r + mg
T = mv²/r
- T is tension in newtons.
- m is mass in kilograms.
- v is speed in meters per second.
- r is radius in meters.
- g is gravitational acceleration.
- θ is measured from the top of the circle.
How to Use This Calculator
- Enter the object mass in kilograms.
- Enter the circular path radius in meters.
- Enter the object speed in meters per second.
- Keep gravity at 9.81 m/s², unless needed otherwise.
- Select the motion scenario you want to analyze.
- For a custom vertical point, enter the angle from the top.
- Press Calculate Tension to view results and the graph.
- Use the CSV or PDF buttons to save the calculation.
Frequently Asked Questions
1. What does this calculator measure?
It calculates the rope or string tension needed to maintain circular motion. It also shows centripetal force, radial acceleration, angular velocity, period, and a tension graph.
2. Why does tension change in a vertical circle?
Gravity helps or opposes the needed inward force. At the top, gravity points inward. At the bottom, gravity points outward from the center, so tension must increase.
3. Why can the calculated tension become negative?
A negative value means the rope would need to push inward, which it cannot do. In that case, the rope goes slack and the circular path cannot stay fully constrained.
4. What angle definition does the tool use?
The calculator measures angle from the top of the circle. Top is 0°, side is 90°, and bottom is 180°.
5. Does this work for flat horizontal motion?
Yes. In a flat horizontal circle, tension supplies the full centripetal force. The calculator switches to T = mv²/r and displays a constant-tension graph.
6. Which units should I enter?
Use kilograms for mass, meters for radius, meters per second for speed, and meters per second squared for gravity. The output tension is given in newtons.
7. What happens if speed doubles?
Tension changes with v², not just v. If speed doubles, the centripetal part becomes four times larger, so total tension rises sharply.
8. Can this replace a full dynamics analysis?
It is excellent for standard circular motion cases. More complex systems with drag, elastic ropes, changing speed, or multiple forces need a broader dynamics model.