Hanging Cords Between Unequal Poles, a Bounce, and a Beam
Where a slack cord touches ground between supports of different heights; rebound and beam sag
Four studies on this leaf concern cords, rebound, and beams. Leonardo states that a slack cord slung between two supports of unequal height touches ground nearer the shorter support, in the same ratio by which the taller contains the shorter, working through the points a, b, c, d, e, h, g. A note on rebound holds that each bounce of a ball rises to one-third the height of the one before. A closing problem asks how far a four-braccio board, which sags one braccio under a weight placed two braccia from a pillar, will sag when that weight is moved to one braccio away.
On this page
Where a slack cord touches the ground (a, b, c, d, e)
The lowest point of the arc of a cord longer than the gap between its supports, hung with its ends at two different heights, touches ground as much closer to the shorter support as the taller receives the height of the shorter into itself. If support a b enters twice into the taller support e d, then the distance between c and b likewise enters twice into d c.
A weight on a slack cord rests off-center (g, h)
A weighty body hung on a cord whose ends are fixed at different heights settles between equal angles, resting nearer one support than the other as the shorter support enters into the taller. It is impossible for such a weight to come to rest over the middle of the span when the supports are unequal, even were it to descend to the center of the earth.
Each bounce rises to one-third the last
Under the note 'Blow and bounce,' Leonardo records that the second bounce of a ball rises to one-third the height of the first, and likewise the third rises to one-third the height of the second.
Problem: deflection of a loaded board between pillars
A weight on a four-braccio board suspended between two pillars makes it sag one braccio at the middle when placed two braccia from a pillar. Leonardo poses the problem: if the same weight is moved to only one braccio from the pillar, how much will the board then sag?