Euler–Eytelwein (capstan) equation

For a tiny rope segment with angle dβ, normal force scales with local tension and tangential friction is at most μ times normal force. Force balance gives dT/T=μ dβ. Integrating over the contact angle yields ln(T1/T2)=μβ and therefore the exponential tension ratio. The relation describes impending slip at cylindrical contact. Clearly identify the friction coefficient, wrap angle and tight/slack sides. Application: Model rope winches, band brakes and capstan friction. The numerical example For μ=0.30, β=π rad and T2=100 N, T1,max=100e^(0.30π)≈256.6 N. Maximum tension difference is 156.6 N. Less wrap or friction moves the allowable ratio toward one anchors the idea in a concrete measurement case.

Use radians because the angle comes from integrating differential angular segments. A half-wrap is β=π, not the number 180. Entering 180 makes the exponent about 57.3 times too large. Also check which side has the greater tension. The relation describes impending slip at cylindrical contact. Clearly identify the friction coefficient, wrap angle and tight/slack sides. The corresponding units are Tensions in N; μ dimensionless; β in rad; μβ dimensionless. For μ=0.30, β=π rad and T2=100 N, T1,max=100e^(0.30π)≈256.6 N. Maximum tension difference is 156.6 N. Less wrap or friction moves the allowable ratio toward one. This checks that the reference quantities are compatible.

At T1/T2=e^(μβ), the ideal rope is at the friction limit. A small extra pull on the tight side causes slip. A resting rope may have a smaller ratio; the equation does not prescribe actual tensions. It only bounds the transferable ratio under its contact assumptions. The relation describes impending slip at cylindrical contact. Clearly identify the friction coefficient, wrap angle and tight/slack sides. Scope: At impending slip with Coulomb friction, segment force balance integrates to T1/T2=e^(μβ). In static contact, T1/T2≤e^(μβ). Assumptions are a flexible massless line, constant friction and cylindrical contact. V-belts need another effective-friction model because of wedge action. Use radians in the exponent. A practical application is Model rope winches, band brakes and capstan friction.

The exponential relation follows by integrating differential force balance along the contact arc. The angle must be inserted as a dimensionless radian value. Putting degrees directly into e^(μβ) changes the exponent by a factor of 180/π and produces a large error. Convert 180° to π rad and check that μβ is dimensionless. The relation describes impending slip at cylindrical contact. Clearly identify the friction coefficient, wrap angle and tight/slack sides. Check the calculation with this case: For μ=0.30, β=π rad and T2=100 N, T1,max=100e^(0.30π)≈256.6 N. Maximum tension difference is 156.6 N. Less wrap or friction moves the allowable ratio toward one. Keep the model limits and units in view.

The limiting relation describes impending slip for a flexible line, constant Coulomb friction and cylindrical contact. In static contact it is an inequality; actual tension must remain below the limit. Elastic stretch, line mass, changing friction, V-grooves and acceleration need extensions. A belt’s wedge action changes effective friction, so the basic capstan value does not fully model belt drive. The relation describes impending slip at cylindrical contact. Clearly identify the friction coefficient, wrap angle and tight/slack sides. The relation describes The capstan relation gives maximum tension ratio for a flexible line wrapped around a drum at impending slip. This helps distinguish its target quantity from neighbouring quantities.