A quarter-circle block with a vertical rectangular end is attached to a balance beam as shown in Fig. 1. Water in the tank puts a hydrostatic pressure force on the block which causes a clockwise moment about the pivot point. This moment is balanced by the counterclockwise moment produced by the weight placed at the end of the balance beam. The purpose of this experiment is to determine the weight, W, needed to balance the beam as a function of the water depth, h.
For a given water depth, h, determine the hydrostatic pressure force, FR = γhcA, on the vertical end of the block. Also determine the point of action of this force, a distance yR – yc below the centroid of the area. Note that the equations for FR and yR – yc are different when the water level is below the end of the block (h < R2 – R1) (partially submerged) than when it is above the end of the block (h > R1 – R2) (fully submerged).
(a) Sketch the problem for a fully immersed block. Find W = f1 (h) symbolically. Then substitute for the constants. Obtain a linear relation for W as a function of h.
(b) Sketch the problem for a partially immersed block. Find W = f2 (h) symbolically. Then substitute for the constants. Obtain a cubic relation for W as a function of h.