Capstan: Coefficient of Friction μ
Known holding tension T₁. Measure the just-slipping T₂ at several wrap angles and recover μ from a fit of ln(T₂/T₁) versus φ.
Goal
Determine μ from T₂ = T₁ e^{μφ}. The slope of ln(T₂/T₁) versus φ (radians) is μ.
Equipment
- Capstan drum
- Known T₁
- Load cell (T₂)
- Wrap-angle scale
Experiment
Theory
Euler–Eytelwein: an infinitesimal wrap dφ gives dT = μ T dφ, so T₂/T₁ = e^{μφ}. The bench hides μ, T₂/T₁ and live T₂. T₁ is known. You set the wrap angle.
Procedure
- Holding tension T₁ is fixed and known. μ is hidden. You only change the wrap angle φ.
- Record. A load cell logs the just-slipping T₂ with small noise. There is no live T₂ or T₂/T₁.
- The notebook converts φ to radians and computes ln(T₂/T₁). Repeat for at least 6 wraps from about 60° to 360°.
- Fit ln(T₂/T₁) versus φ (rad); the slope is μ. Compare with the reference.
Conclusion
The fitted μ agrees with the hidden friction coefficient. Main uncertainties: load-cell noise and treating the rope as massless with uniform μ.