# At which point does the pressure of the water act on a dam?

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.

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

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 $$z=h$$.

$$d \tau$$=$$F$$×$$r$$

=$$\rho g (h-y)(dy)$$×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