Intrinsic equation of a catenary (relation between arc length and slope angle) For a catenary defined by parameter c, which of the following is the intrinsic (s–ψ) relation linking arc length S to the angle ψ that the tangent makes with the horizontal?

Introduction / Context:
A uniform flexible cable hanging under its own weight forms a catenary. Different equivalent forms exist: Cartesian (x–y), parametric, and intrinsic (relating arc length to slope angle). Recognizing these is important in cable and suspension-bridge analysis.

Given Data / Assumptions:

• Standard catenary with parameter c (ratio T0/w where T0 is horizontal tension and w is weight per unit length).
• ψ is the angle between the tangent and the horizontal.
• Use known identities for catenary geometry and statics.

Concept / Approach:

In intrinsic form, the relation between arc length S measured from the lowest point and the slope angle ψ satisfies S = c tan ψ. Cartesian form through the lowest point is y = c cosh(x/c). Other expressions listed mix forms or variables incorrectly.

Step-by-Step Solution:

Verification / Alternative check:

Differential geometry of a catenary gives dS = √(1 + (dy/dx)^2) dx; using dy/dx = sinh(x/c), integrate to get S = c sinh(x/c); combine with tan ψ = sinh(x/c) to obtain S = c tan ψ.

Why Other Options Are Wrong:

(b) is the correct Cartesian equation, not intrinsic; (c) and (d) assign ψ incorrectly as the argument to hyperbolic functions of y; (e) is another form for S but in terms of x, not ψ.

Common Pitfalls:

Confusing ψ with x/c; mixing up sinh and cosh roles; forgetting that intrinsic form eliminates x and y in favor of S and ψ.

Final Answer:

S = c tan ψ