Question

Force on a dam Find the total force on the face of a semicircular dam with a radius of 20 m when its reservoir is full of water. The diameter of the semicircle is the top of the dam.

Accepted Answer

Identify the physical setup: The dam face is a vertical semicircle with radius $$r = 20 m$$, and the water exerts pressure on it. The pressure at a depth $$h in a $$fluid is given by $$p = ho g h$$, where $$ho is $$the density of water and $$g is $$the acceleration due to gravity. Set up a coordinate system: Let the vertical axis $$y$$ measure depth from the water surface downward, with $$y=0 at $$the top (diameter) of the semicircle and $$y$$ increasing downward to $$y=20 m at $$the bottom of the dam. Express the width of the dam at depth $$y$$: Since the dam face is a semicircle, the horizontal width at depth $$y$$ corresponds to the length of a horizontal chord of the semicircle. Using the circle equation $$x^2$$ + $$y^2$$ = $$r^2$$, the half-width at depth $$y is$$ $$x = \sqrt{r^2$ - y^2$}$$, so the full width is $$2\sqrt{r^2$ - y^2$}. $$Write the differential force $$dF on a $$thin horizontal strip of thickness $$dy at $$depth $$y$$: The pressure at depth $$y is$$ $$p = ho g y$$, and the area of the strip is $$dA = 	ext{width} 	imes dy = 2\sqrt{r^2$ - y^2$} \, dy. $$Therefore, $$dF = p \, dA = ho g y \cdot 2\sqrt{r^2$ - y^2$} \, dy. $$Integrate the differential force over the depth of the dam from $$y=0 to$$ $$y=r to $$find the total force: $$F = \$$int_0$$^r 2 ho g y \sqrt{r^2$ - y^2$} \, dy. $$Set up this integral and prepare to evaluate it to find the total force on the dam face.

Force on a dam Find the total force on the face of a semicircular dam with a radius of 20 m when its reservoir is full of water. The diameter of the semicircle is the top of the dam.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Hydrostatic Pressure Distribution

Force on a Curved Surface

Geometry of a Semicircle and Coordinate Setup