In 1696, the first calculus textbook was published by the Marquis de l'Hôpital. The following problem is a simplified version of a problem from this text.
In Figure 2 below, we have two ropes attached to a ceiling, 3m apart. A pulley is attached to the rope on
the left, which has a length of Im. The other rope has length 5m, and it passes through the pulley and has a
weight attached to the end of it. When the weight is released, the ropes and pulley rearrange themselves so
that the diptance from the weight to the ceiling is maximized. As a simplifying assumption in this problem,
you may assume that the radius of the pulley is small enough to be negligiblo.
(a) Create an equation which gives the distance from the weight to the ceiling as a function of 0, the angle
that the Im rope makes with the ceiling.
(6) Use this to determine the value of O for which this distance is maximized.
(e) Determine the horizontal position that the weight will assume once the ropes and pulleys have adjusted
themselves to an equilibrium state.
3 meters
1 meter
Weight