Solve capstan holding tension, load tension, friction coefficient, wrap angle, required turns, and slip margin with automatic unit conversion.
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Capstan Equation Formula
The capstan equation relates the tension on one side of a rope, belt, cable, or line wrapped around a post, drum, pulley, or capstan to the tension on the other side.
Rearranged forms used by the calculator:
- Tin = holding-side (smaller) tension for the selected impending-slip direction
- Tout = load-side (larger) tension for the selected impending-slip direction
- ฮผ = coefficient of friction between the line and the surface
- ฮธ = wrap angle in radians
- e = Eulerโs number, approximately 2.71828
Choose the quantity in the Solve for selector, then complete the visible inputs. The calculator solves load-side tension, holding-side tension, friction coefficient, or required wrap. The equation gives the limiting tension ratio at impending slip, as discussed in MITโs capstan-equation lecture. Reversing impending motion exchanges the load and holding roles. Required wrap is reported in turns, with radians and degrees also shown.
Tension inputs share the selected newton, kilonewton, pound-force, or kilogram-force unit. Wrap angle accepts degrees, radians, or turns; the equation uses radians.
Wrap Angle and Friction Reference Values
The wrap angle is the total contact angle between the line and the surface. More wrap angle increases the possible tension ratio exponentially.
| Wrap | Degrees | Radians | Description |
|---|---|---|---|
| 1/4 turn | 90ยฐ | 1.5708 | Line contacts one quarter of the cylinder |
| 1/2 turn | 180ยฐ | 3.1416 | Line wraps halfway around |
| 1 turn | 360ยฐ | 6.2832 | One complete wrap |
| 2 turns | 720ยฐ | 12.5664 | Two complete wraps |
| Illustrative scenario | Assumed ฮผ range | Notes |
|---|---|---|
| Low coefficient | 0.10 to 0.20 | Illustration only; not a measured material value |
| Moderate coefficient | 0.30 to 0.50 | Use a measured value for the actual contact pair |
| Intermediate coefficient | 0.25 to 0.45 | No universal roughness-to-friction mapping is assumed |
| Higher coefficient | 0.40 to 0.80 | Check the actual belt and pulley specification |
Examples
Example 1: Calculate outgoing tension
You have an incoming tension of 100 N, a coefficient of friction of 0.30, and a wrap angle of 180ยฐ.
Convert the angle to radians:
Apply the capstan equation:
The outgoing tension is approximately 256.63 N.
Example 2: Calculate coefficient of friction
You have an incoming tension of 50 N, an outgoing tension of 200 N, and a wrap angle of 360ยฐ.
Convert the angle to radians:
Solve for ฮผ:
The coefficient of friction is approximately 0.2206.
FAQ
Does the wrap angle have to be in radians?
Yes, the capstan equation uses radians. If you enter degrees, the calculator converts degrees to radians before applying the formula. For example, 180ยฐ is 3.1416 radians, and 360ยฐ is 6.2832 radians.
Why does tension increase so much with extra wraps?
The equation uses an exponential term, eฮผฮธ. That means the tension ratio does not increase linearly. Adding more wrap angle can greatly increase the holding effect, especially when the coefficient of friction is high.
Can the coefficient of friction be negative?
For the forward tension modes, ฮผ may be zero or positive. A unique inverse wrap solution requires positive ฮผ. A negative value would imply the wrapped contact reduces the tension ratio in the opposite direction, which does not match the usual friction model. If the calculated ฮผ is negative, check whether the incoming and outgoing tensions were entered in the correct fields.
