Tension of a Soft Spring in Contact with a Cylinder

It is possible for a relatively small force, applied to one end of a rope, to support a much larger force if the rope is merely wrapped a few times around a post. This setup, called a capstan, has been discussed in a number of papers in this journal and elsewhere.^1–3 If we wrap a cord around a rough curved surface, T₁ and T₂ are the tensions in the cord on the two sides of the surface (see Fig. 1). At the point the cord is about to slip, the theory predicts¹ T₂ = T₁e^μθ, (1) where μ is the coefficient of static friction between the cord and surface. The amount of wrapping is indicated by the angle θ measured in radians. A common experiment is to measure T₁ and T₂ for various values of θ and to obtain the coefficient of friction μ from the slope of a plot of ln (T₂/T₁) versus θ. [Note that Eq. (1) implies ln (T₂/T₁) = μθ.] The tension in the string varies exponentially with position along the cylinder. It would be difficult to measure that variation using the experiment represented by Fig. 1. However, if the string is replaced by a tightly wound spring, the tension in the spring at any point can be found from the separation between adjacent coils (greater separation means greater tension). Among the capstan‐related experiments described by Levin³ is one in which a spring with unequal weights connected to its ends is draped over a rough‐surfaced horizontal cylinder (Fig. 2). With this arrangement, Levin is able to demonstrate qualitatively the variation in the tension in the spring. We describe here a similar experiment in which quantitative measurements of the tension variation in the spring can be obtained. From those measurements, the coefficient of static friction between the spring and the cylinder can be calculated.