Demonstration of the weight of the arched cord
Locating the center of gravity and the line of force of a hanging cord as an arc of a circle
This page demonstrates the weight carried by an arched cord whose ends hang level, showing that such a cord always describes an arc of a circle with its center of gravity at the midpoint of its length. Leonardo gives a construction: find the center h of the whole circle, draw the radius h a, raise perpendiculars a n and c e at the ends of the arc, and use these as real levers against the counter-lever a d carrying the weight's pendant d L. A marginal rule explains his drawing convention, that doubled lines mark real members and single lines mark potential lines, with some figures purely one, purely the other, or composite; a large arched-cord figure and a smaller loaded frame appear at the right.
On this page
An arched cord in equilibrium describes an arc of a circle
A cord that curves with its ends level always describes part of the circumference of a circle, and its center of gravity is always found at the middle of its length. The line bearing the weight of the whole arched cord is straight and starts where the circular line separates from its tangent.
Constructing the circle's center to find the line of force
Taking the arc a b c as part of a circle, find the center h of the whole circle, draw the straight line h a to get the radius, and at its end raise the perpendicular a n; do likewise at the opposite end with line c e. Join perpendicular e a as the real lever to the counter-lever a d, and to d attach the weight's pendant d L.
Convention: doubled lines are real, single lines potential
All figures with doubled lines are to be understood as real members, and those with single lines as potential lines. Among the figures there are simple potential ones, simple real ones, and composites of real and potential.