Movable Pulley on a Hyperbolic Trajectory and Nonlinear Tension

Problem Statement

Set up an $x$x–$y$y coordinate plane. The positive $y$y-direction is vertically upward; the positive $x$x-direction is horizontally to the right. The magnitude of gravitational acceleration is $g$g.

A light, inextensible string has one end fixed to point $\text{A}(0, H)$A(0,H) on the ceiling. A smooth fixed pulley of negligible size and mass is attached to point $\text{B}(L, 0)$B(L,0) on a wall.

A smooth movable pulley of mass $M$M (negligible size) is threaded onto the string, which then runs over the fixed pulley at $\text{B}$B. The free end of the string is pulled slowly with force $F$F.

The movable pulley maintains quasi-static force balance as it moves in the $x$x–$y$y plane. (As the string is pulled, the movable pulley traces a smooth curve moving upward and to the right toward pulley $\text{B}$B.)

Find the magnitude of force $F$F at the instant when the $x$x-coordinate of the movable pulley is $x = \frac{3}{4}L$x=43​L.

Pulley friction and string mass are negligible. The geometry is ideal (no interference between string sections).

Constraints

• $H = 2.0\,\text{m}$H=2.0m
• $L = 3.0\,\text{m}$L=3.0m
• $M = 16\,\text{kg}$M=16kg
• $g = 9.8\,\text{m/s}^2$g=9.8m/s2

Input Format

Find the magnitude of force $F$F in N and give the answer as a positive integer.