§ 10. Statics of Inextensible Chains.—The theory of bodies or structures which are deformable in their smallest parts belongs properly to elasticity (q.v.). The case of inextensible strings or chains is, however, so simple that it is generally included in expositions of pure statics.

It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element δs must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4).

Fig. 54.

We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= δs) in the directions of the tangent and normal respectively. If T, T + δT be the tensions at P, Q, and δψ be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + δT) cos ψ − T, or δT, ultimately; whilst for the component along the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius of curvature.

Suppose, for example, that we have a light string stretched over a smooth curve; and let Rδs denote the normal pressure (outwards from the centre of curvature) on δs. The two resolutions give δT = 0, Tδψ = Rδs, or

T = const.,   R = T/ρ.

(1)

The tension is constant, and the pressure per unit length varies as the curvature.

Next suppose that the curve is “rough”; and let Fδs be the tangential force of friction on δs. We have δT ± Fδs = 0, Tδψ = Rδs, where the upper or lower sign is to be taken according to the sense in which F acts. We assume that in limiting equilibrium we have F = μR, everywhere, where μ is the coefficient of friction. If the string be on the point of slipping in the direction in which ψ increases, the lower sign is to be taken; hence δT = Fδs = μTδψ, whence

T = T0 eμψ,