How it works
When a rope wraps around a fixed cylinder, friction between the two grows exponentially with the angle of wrap. The tension on the load side can be far larger than the holding force on the tail: T_load = T_hold × e^(μθ), where θ is the total wrap angle in radians and μ is the friction coefficient.
One full wrap is 2π radians. The calculator solves the equation two ways: how many wraps a given holding force needs, and how much holding force your chosen number of wraps provides. The friction coefficient is the weak link in the answer, and it changes with rope, surface, and wear.
Worked example
A 500 lb rope tension held with 50 lb of hand force at μ = 0.3 (example value):
- Wraps needed = ln(500 ÷ 50) ÷ (2π × 0.3) = 1.22
- Holding force with 2 wraps = 500 ÷ e^(0.3 × 2π × 2) = 11.5 lb
| Input | Value |
|---|---|
| Rope tension on the load side | 500 lbf |
| Holding force you can safely apply | 50 lbf |
| Friction coefficient of rope on the device | 0.3 μ |
| Wraps you plan to use | 2 wraps |
| Result | Value |
|---|---|
| Wraps needed to hold with your force | 1.22 wraps |
| Holding force with your planned wraps | 11.5 lbf |
| Load ÷ holding force with your wraps | 43.4 × |
Assumptions and limits
- The friction coefficient is a single value you supply; the calculator does not know your rope, device, or its condition.
- The capstan equation is for a rope that is slipping on a fixed cylinder at steady speed. Real lowering involves changing loads, heat, and glazing that lower friction.
- It gives no strength rating for the rope, the device, or the tree.
Common questions
How many wraps do I need to hold a load?
Enter the tension, your holding force, and the friction coefficient; the calculator gives the wraps. Each wrap multiplies your holding power by e^(2πμ).
What friction coefficient should I use?
It depends on the rope and device and changes with wear, moisture, and heat. Use manufacturer data or measure it; the 0.3 shown is only an example.
Why does the holding force drop so fast with more wraps?
Because the effect is exponential in the wrap angle, a couple of extra wraps can cut the holding force by an order of magnitude.
Can I rely on this for lowering a heavy limb?
No. Use it to understand the friction only; follow your device’s instructions, rated equipment, and a qualified person’s judgment.
Sources
- Capstan equation (Euler–Eytelwein formula): T_load = T_hold × e^(μθ), standard in mechanics texts.
- Wikipedia: Capstan equation.
- Read them: Capstan equation (Wikipedia)
Updated 2026-09-30