Vertical Circular Motion Calculator

This tool assumes a point mass and a rigid radius. Air drag and friction are ignored.

Example Data Table

Formula Used

How to Use This Calculator

1. Select a calculation mode that matches what you know.
2. Enter the radius and the angle from the bottom.
3. Provide mass when forces or energies are required.
4. For energy mode, enter the speed at the bottom.
5. For tension mode, enter the desired radial force.
6. Click Calculate to view results above the form.
7. Use the download buttons to export results as CSV or PDF.

Vertical Circular Motion Notes

1) Overview

Vertical circular motion is motion on a fixed-radius circle while gravity acts downward. Speed, tension, and required centripetal acceleration vary with position because the height changes. This calculator assumes a point mass and an ideal path with no friction or drag.

2) Angle and height

The angle θ is measured from the bottom: 0° bottom, 90° side, 180° top. The height above the bottom is h = r(1 − cosθ). With r = 0.50 m, the side is 0.50 m higher and the top is 1.00 m higher.

3) Centripetal acceleration

Centripetal acceleration is a_c = v²/r toward the center. The displayed g‑force is a_c/g. Example: v = 5.0 m/s, r = 0.50 m gives a_c = 50 m/s², about 5.10 g.

4) Tension or normal force

Using inward as positive, the radial balance is T = m v²/r + m g cosθ. At the bottom, cosθ = 1 so tension is highest. At the top, cosθ = −1 so gravity reduces tension and contact can be lost if T becomes negative.

5) Energy-based speed

If bottom speed is v₀, the ideal speed at angle θ follows v² = v₀² − 2gr(1 − cosθ). When v² drops below zero, the bottom speed is insufficient to reach that angle without extra energy input.

6) Minimum complete loop

For just-maintained contact at the top, set T = 0 at 180°, giving v_top = √(gr). Combining with the 2r height rise gives v_bottom = √(5gr). With g = 9.80665 m/s² and r = 0.50 m, v_top ≈ 2.214 m/s and v_bottom ≈ 4.950 m/s.

7) Practical design data

Loads scale strongly with speed: doubling v quadruples v²/r, so small speed increases can raise tension sharply. Real loops lose energy, so use safety margins above the minimum. The tension mode is useful for specifying a target radial load, then checking whether it is reasonable at the chosen angle in practice.

FAQs

1) What does “minimum speed for contact” mean?

It is the smallest speed at a chosen angle that keeps tension or normal force non‑negative. Above 90°, too little speed can make the mass lose contact because gravity cannot supply the required inward centripetal force.

2) Why can the speed become imaginary in energy mode?

Energy mode uses v² = v₀² − 2gr(1 − cosθ). If v₀ is too small to climb to that height, v² becomes negative. The tool reports this as insufficient bottom speed.

3) Does mass affect the minimum complete-loop speed?

In the ideal model, the minimum loop speeds depend on g and r, not mass. Mass cancels in the top contact condition and in the energy step from bottom to top.

4) Why is tension usually highest at the bottom?

At the bottom, centripetal demand m v²/r points upward and weight adds to the needed inward force, giving T = m v²/r + mg. Higher tension is common even if the speed is similar elsewhere.

5) Can I use this for a car in a loop-the-loop?

It can estimate forces for a simplified point-mass path. Real vehicles have suspension travel, aerodynamic drag, and changing tire contact, so treat results as rough and apply larger safety factors.

6) Which unit conversions are included?

Mass supports kg, g, and lb. Radius supports m, cm, mm, and ft. Speed supports m/s, km/h, and mph. Tension supports N, kN, and lbf, with results reported consistently.