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For this problem,

enter image description here

So they assume that the pressure at the top of the dam is zero instead of 1 atm. Therefore I wanted to try to take atmospheric pressure into account,

enter image description here

Do you please know how to get P_0A/2 so that the P_0A/2 terms can cancel to get their hydrostatic force? I assume that the atmospheric pressure is the same at the top and bottom since the height of the dam is negligible to the scale of the atmosphere.

Thanks for any help!

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First of all, I wouldn't suggest using an average of forces, as they can be wrong in many cases. Calculus way though it's hard if correctly done is a very concise way to solve problems.

Let us take a strip of the dam of breadth $dy$ and length $l$ and y depth (from the surface). So force acting would be:

$$dF=(\rho gy)(ldy)$$ Integrating: $$F=\frac{\rho g hA}{2}$$

Interesting to note that I did not take $P_{atm}$ is because both water and dam surface is open to the atmosphere, so they are automatically canceled.

Edit:

Taking, $P_{atm}$ into equations. Firstly we see that P is acting on water surface also so, $$F=\frac{\rho g hA}{2}+P_0A$$ Now also P acts on dam's other side on opposite direction, which is: $$F_1=P_0 A$$ $$F_{net}=F-F_1$$

EDIt-2

So lets say our water reservoir of dam is open to atomosphere. We need to calculate pressure from atomosphere only and corresponding force on dam.

$$dF_{atomosphere}=(P_0)(ldx)$$

Note that $P_o$ is not function of depth like we did for water. Integrating: $$F_{atomosphere}=P_0 (lx)$$ $$F_{atomosphere}=P_0 (A)$$

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  • $\begingroup$ Thanks for your answer! Could you please show me how to do it if I do take Patm into account? Many thanks. $\endgroup$
    – Quantum guy
    Nov 21, 2022 at 20:05
  • $\begingroup$ @Quantumguy Added the new edits. If you are satisfied with answer, you can press the tick symbol under vote buttons. Tell me if other doubts persist. $\endgroup$
    – Kshitij Kumar
    Nov 22, 2022 at 5:37
  • $\begingroup$ Thanks for your edits and comment! Sorry where did your get the first equation (F=ρghA2+P0A) from? Thank you! $\endgroup$
    – Quantum guy
    Nov 22, 2022 at 5:41
  • $\begingroup$ Do you please know why it is not F = (P_0 + pgh)A/2? $\endgroup$
    – Quantum guy
    Nov 22, 2022 at 5:53
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    $\begingroup$ @Quantumguy Welcome. $\endgroup$
    – Kshitij Kumar
    Nov 22, 2022 at 7:19

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