Question

Force on a triangular plate A plate shaped like an isosceles triangle with a height of 1 m is placed on a vertical wall 1 m below the surface of a pool filled with water (see figure). Compute the force on the plate.

Accepted Answer

Identify the physical context: The force on the triangular plate submerged in water is due to the hydrostatic pressure exerted by the water on the plate's surface. Recall the hydrostatic pressure formula: The pressure at a depth $$h in a $$fluid of density $$ho$$ under gravity $$g is $$given by $$P = ho g h. $$Here, $$h is $$the vertical distance below the water surface. Set up a coordinate system: Let $$y$$ measure the vertical distance from the water surface downward. The top of the plate is at $$y=1 m $$and the bottom at $$y=2 m. $$Express the width of the triangular plate at depth $$y$$: Since the plate is an isosceles triangle with height 1 m and base 1 m, the width at depth $$y ($$where $$y$$ ranges from 1 to 2) changes linearly. The width $$w(y)$$ can be expressed as $$w(y) = 2(y - 1)$$ meters, because at $$y=1$$ the width is 0 and at $$y=2$$ the width is 1 m. Calculate the force by integrating pressure over the area: The differential force $$dF on a $$thin horizontal strip of thickness $$dy at $$depth $$y is$$ $$dF = P(y) 	imes w(y) 	imes dy = ho g y 	imes w(y) 	imes dy. $$Integrate $$dF$$ from $$y=1 to$$ $$y=2 to $$find the total force on the plate.

Force on a triangular plate A plate shaped like an isosceles triangle with a height of 1 m is placed on a vertical wall 1 m below the surface of a pool filled with water (see figure). Compute the force on the plate.

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Key Concepts

Hydrostatic Pressure

Force on a Submerged Surface

Geometry of the Triangular Plate