A triangle with base 3 m and height 5 m is submerged vertically in water so that the tip is even with the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it.
This is a WebAssign problem - I've been following the Master It example which shows a similar problem with different numbers, but I can't figure out how they went from
F=3/5pg[x^3/x] from x=0-5 to 588,000. They completely skip this step, and I feel as though its an interesting one to skip...Plus 588,000 isn't the correct answer. Any help would be great, thank you!
$\endgroup$1 Answer
$\begingroup$Consider integrating the hydrostatic force from the top of the triangle to the bottom. For a thin horizontal slice of the triangle the force is $\delta F = \rho g x w \space \delta x$ At depth $x$, the width of the triangle is $w = 3/5x$.
$F = \int_0^5 \rho g xw dx = \frac{3}{5}\rho g \int_0^5 x^2 dx = \frac{3}{5}\rho g \frac{x^3}{3} \Big|_0^5 = \frac{1}{5}\rho g (25 - 0) = 5\rho g \approx 49 kN$
Always give your answer with units!
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