Questions: Friction in Belt and Rope Systems

Question 1

A rope with μ = 0.3 is wrapped once around a bollard (β = 2π), giving T₂/T₁ ≈ 6.6. A sailor adds a second complete wrap (β = 4π). What is the approximate new tension ratio?

• A. About 13 — the ratio doubles when the wrap angle doubles
• B. About 20 — the ratio increases proportionally to wrap angle
• C. About 43 — the ratio approximately squares because the relationship is exponential
• D. Still about 6.6 — the ratio is determined by μ alone, not by wrap angle

Answer: (C)

T₂/T₁ = e^(μβ). For β = 2π: e^(0.3 × 2π) = e^1.885 ≈ 6.59. For β = 4π: e^(0.3 × 4π) = e^3.77 ≈ 43.4. Doubling β doubles the exponent, which squares the ratio: 6.59² ≈ 43.4. The relationship is exponential, so each added wrap multiplies the holding capacity by a factor equal to the previous ratio — not adds to it. This is the capstan's remarkable mechanical advantage: adding one wrap to an existing four can multiply holding force by another factor of 6.

Question 2

What is the physical reason the Capstan equation is exponential rather than linear in wrap angle?

• A. The belt material stiffens as it wraps further, increasing the effective friction coefficient
• B. Each increment of wrap generates a normal force proportional to the local tension, which itself grows as friction accumulates — a self-reinforcing feedback
• C. The normal force is uniformly distributed along the contact arc, independent of tension magnitude
• D. The coefficient of friction μ increases with wrap angle due to frictional heat generation

Answer: (B)

For a short rope segment subtending dβ, the inward (centripetal) component of the tension forces is dN = T dβ. The friction on that segment is μ dN = μT dβ, and it adds to the tension. But T itself has been growing from accumulated friction upstream. So each successive segment has a larger T, generates a larger dN, and adds more friction — a self-reinforcing process. Setting up dT = μT dβ and integrating gives the exponential T₂ = T₁ e^(μβ). The exponential arises because the source of friction (normal force) is proportional to the effect (tension growth).

Question 3

Doubling the wrap angle of a rope around a cylinder doubles the maximum holdable tension ratio T₂/T₁.

• T. True
• F. False

Answer: False

Doubling β doubles the exponent in e^(μβ), which squares the ratio. If β₁ gives ratio R₁ = e^(μβ₁), then β₂ = 2β₁ gives R₂ = e^(2μβ₁) = R₁². For example, R₁ ≈ 6.6 becomes R₂ ≈ 43. The relationship is exponential, not linear — this is the entire point of the capstan's dramatic mechanical advantage. Many students assume linearity (doubling β → doubling ratio) and dramatically underestimate the effect of additional wraps.

Question 4

The Capstan equation T₂ = T₁ e^(μβ) gives the maximum tension ratio before slip; at lower applied forces, the actual ratio can be any value from 1 up to this limit.

• T. True
• F. False

Answer: True

The equation is derived by setting friction to its maximum value μN at every point along the wrap — the condition for impending slip. If the applied force T₂ is less than T₁ e^(μβ), static friction has not reached its limit and the system is in static equilibrium. The actual tension ratio depends on the applied loads, not just μ and β. The Capstan equation specifies the ceiling: once T₂/T₁ > e^(μβ), the rope slips regardless of how it got there.

Question 5

A sailor wraps a dock line around a bollard to hold a large boat. Explain, using the Capstan equation, why adding even one more complete wrap dramatically increases the holding force.

Model answer: By T₂ = T₁ e^(μβ), the ratio of holding tension to effort is exponential in wrap angle β. Adding one complete wrap increases β by 2π, multiplying the exponent by that amount. If the existing ratio after n wraps is R = e^(μ × 2πn), adding one more wrap gives R × e^(2πμ) — the new ratio is the old one multiplied by e^(2πμ), which for typical μ = 0.3 is a factor of about 6.6. So a sailor who could hold 100 N of boat force with one wrap can hold 660 N with two, 4,360 N with three — each wrap multiplying, not adding to, the mechanical advantage.

This exponential amplification is why rope-and-bollard systems, rock-climbing belay devices, and capstan winches all work on the same principle: friction accumulates along the wrap because the friction force at each point depends on the local tension, which has already been amplified by friction upstream. The practical implication is that a small person can hold an enormous load with a few wraps, as long as the rope doesn't slip.