Final answer:
To find the hydrostatic force against one side of a submerged equilateral triangular plate, we integrate the water pressure over the surface area of the plate. As water pressure increases with depth, we evaluate the integral for the pressure across the triangle's height, considering the triangular geometry and linear decrease of base with height.
Step-by-step explanation:
To calculate the hydrostatic force against one side of an equilateral triangular plate submerged in water, we need to integrate the pressure over the area of the side of the plate. The pressure at a depth h is given by P = pgh, where p is the density of water, g is the acceleration due to gravity, and h is the depth from the surface. For an equilateral triangle submerged vertically with the base at the surface, h will vary linearly from 0 at the top to the height of the triangle at the bottom. The height of an equilateral triangle with side s is given by h = (s*sqrt(3))/2. To find total force, we will integrate the pressure over the triangular area.
Thus, the hydrostatic force F can be expressed as:
F = ∫ pgh dA
Where dA is an infinitesimally small area of the triangle, and the integral is taken over the entire surface area of the submerged side. To evaluate this integral, we divide the triangle into horizontal strips, each with area dA = base * dh, and integrate from 0 to the height of the triangle. Since the base of each strip decreases linearly with height, we can express it in terms of h and perform the integration.
The integral then becomes:
F = ∫0(8*sqrt(3))/2 p g ((8 - (2h/sqrt(3))) dh
Upon evaluating this integral, we will obtain the total hydrostatic force exerted on one side of the plate.