This tool uses the classic capstan (belt friction) equation
T₂ = T₁ · e^{μθ} to estimate the tension amplification
on a winch drum or capstan. Use consistent units for all forces
(e.g., all in lbf or all in N).
Capstan Winch Equation – Quick Cheat Sheet
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Core equation:
T₂ = T₁ · e^{μθ}
T₁ = smaller tension (holding / tail line), T₂ = larger tension (load side),
μ = friction coefficient, θ = wrap angle in radians.
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Wrap angle for w wraps (full turns) is
θ = 2π · w.
Example: 3 wraps ⇒ θ ≈ 2π · 3 ≈ 18.85 rad.
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Tension ratio:
T₂ / T₁ = e^{μθ}.
Once you know μ and wraps, the ratio is fixed. You can scale up/down for any units.
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Solving directions:
- Given Thold (T₁):
Tload = Thold · e^{μθ}.
- Given Tload (T₂):
Thold = Tload / e^{μθ}.
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Important note on units: the equation is dimensionless, so as long as you
use the same unit for both tensions (all lbf or all N, etc.), the math is valid.
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Typical μ ranges (very approximate, static friction):
- Synthetic rope on steel:
μ ≈ 0.20–0.35
- Dry fiber rope on steel: similar or slightly higher, depending on condition.
- Rubber on steel: often higher than
0.5 (varies widely).
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Engineering caution: this is a simplified static model. Real systems
must account for dynamic loading, shock, wear, groove shape, lubrication, and safety
factors. Do not use this calculator alone to design life-safety systems.
Quick example: μ = 0.25, 3 wraps ⇒ θ ≈ 18.85 rad.
Tension ratio ≈ e^{0.25 · 18.85} ≈ 111:1.
So only ~100 lbf holding force can theoretically stabilize ~11,000 lbf on the load side
under ideal conditions.
