Suspended Cable Under Self-Weight – Identify the Correct Curve Shape A perfectly flexible, inextensible cable supporting only its own weight (uniformly distributed along its length) hangs freely between two supports. The resulting shape of the cable is:

Introduction / Context:
Recognizing the natural curve of a cable under self-weight is essential in bridge engineering (suspension systems), power-line design, and structural form-finding. The equilibrium of a flexible line under a continuous gravity load leads to a well-known transcendental curve called the catenary.

Given Data / Assumptions:

• Cable is perfectly flexible and inextensible.
• Only load is its own weight (uniform per unit length measured along the cable).
• No uniformly distributed load per horizontal span is assumed here.

Concept / Approach:

For a cable whose load is uniformly distributed along its length, the differential equilibrium leads to the catenary equation: y = a cosh(x/a) − a, where a = H/w, H is horizontal tension and w is weight per unit length. If the load is instead uniform per horizontal projection (as for a level deck with hangers), the cable takes a parabolic shape. Distinguishing these loading modes is the key concept.

Step-by-Step Solution:

Verification / Alternative check:

Compare with the parabola case (uniform load per horizontal span). Real cables supporting their own weight only match the catenary closely; when they carry a uniform deck load, the profile becomes nearly parabolic over typical spans.

Why Other Options Are Wrong:

(a) Parabola corresponds to uniform load per horizontal projection, not self-weight along length. (b) Circle and (d) ellipse are not solutions of the governing equilibrium. (e) A straight line is impossible in equilibrium with gravity; 'small sag' parabolic approximations are only approximations, not the exact curve for self-weight.

Common Pitfalls:

Confusing uniform deck load with self-weight distribution; assuming parabola for all suspended curves.

Final Answer:

Catenary