A control system allows liquid to drain when its level rises sufficiently. A weightless, hollow sphere of radius R is connected via cable (free to slide frictionlessly through a seal, crosshatchd) to the tip of a rectangleshaped gate (shaded), which it holds up against a stop. This stop is a distance a from a frictionless hinge and the gate is of a width w into the paper. (The total gate length is a + b, but only a length a is hydrostatically loaded.) Neglecting the weight of the gate and assuming that the sphere is always completely submerged, determine the height of the liquid h at which the gate will just start to open

8 seal R (10 pts) A control system allows liquid to drain when its level rises sufficiently (Fig. 5). A weightless, hollow sphere of radius R is connected via cable (free to slide frictionlessly through a seal, cross-hatchd) to the tip of a rectangle-shaped gate (shaded), which it holds up against a stop. This stop is a distance a from a frictionless hinge and the gate is of a width w "into the paper". (The total gate length is a +b, but only a length a is hydrostatically loaded.) Neglecting the weight of the gate and assuming that the sphere is always completely submerged, determine the height of the liquid h at which the gate will just start to open. liquid h stop cable - a-t-b- Figure 5: Drain control