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I'm currently working on a problem. I have a dam with height h, and width w. As part of the problem I need the torque resulting from pressing against the dam. I calculated the amount of force applied as follows:

$$ p(z) = \rho \cdot g \cdot z \\ dF = \rho(z)\cdot dA = \rho gwz \cdot dz \\ F = \int_0^h\rho gw z \cdot dz = \frac 12 \rho gwh^2 $$

For the torque, I need the lever arm, for which I need to know where this force acts on the wall, but I'm kind of stuck there. Any help would be greatly appreciated!

Edit: This is the situation I'm dealing with: The water is on the left and I need the torque in regard to axis B.

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  • $\begingroup$ about what point are you trying to compute the torque? The base of the dam? $\endgroup$
    – mike stone
    Commented Jan 9, 2021 at 21:47
  • $\begingroup$ I edited the post, take a look at the picture. $\endgroup$
    – maibrl
    Commented Jan 9, 2021 at 21:54

2 Answers 2

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The pressure force is distributed across the whole dam wall. To find the torque about the bottom of the wall you need to integrate $z \space dF$ from $z=0$ to $z=h$.

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$d \tau$=$F$×$r$

=$\rho g (h-y)(dy)$×y

On integrating and putting limits from 0 to h

$\tau$= $\rho$g$h^3/6$ I.e h/6 from bottom

Extra

For calculating effective distance use torque

$d \tau$=$F$×$r$

=$\rho g y(dy)$×y

$\int \tau$ =$\int$$\rho g y^2(dy)$

$\tau$=$\rho$g$h^3/3$


This shows the effective distance is h/3 from point of application i.e from top

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