Chapter 3.P, Problem 21P

One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d>r), a rope of length $l$ is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. (See the figure.) As 1’Hospital argued, this happens when the distance $\left|ED\right|$ is maximized. Show that when the system reaches equilibrium, the value of x is

$\frac{r}{4d}\left(r+\sqrt{r^2+8d^2}\right)$

Notice that this expression is independent of both W and $l$.