Sag (central dip) of a lightly dipped trolley wire between poles: A trolley wire weighs 1 kg per metre (uniform) and spans 20 m between poles. If the horizontal tension is 1000 kgf, estimate the central dip (assume small sag, parabolic approximation).

Introduction / Context:Light conductors and trolley wires often have small sag. For small sags, the catenary may be approximated by a parabola. The central dip depends on span, weight per unit length, and the horizontal component of tension.

Given Data / Assumptions:

• Uniform weight per horizontal length w = 1 kgf/m.
• Span L = 20 m (level supports).
• Horizontal tension H = 1000 kgf (constant).
• Small sag → parabolic formula valid.

Concept / Approach:For a parabolic cable with small sag, central dip y at midspan is given by y = w * L^2 / (8 * H). Units are consistent when w and H are in kgf and L in metres; result y is in metres.

Step-by-Step Solution:

Verification / Alternative check:If the tension doubled, sag halves, consistent with inverse dependence on H. If span doubled, sag increases fourfold, consistent with L^2 dependence.

Why Other Options Are Wrong:

• 2–4 cm: underestimate; they would require a higher tension or shorter span.
• 8 cm: overestimates sag for the given high tension.

Common Pitfalls:Using total tension instead of its horizontal component; mixing N and kgf inconsistently; forgetting the factor of 8.

Final Answer:5 cm