Homework Help: Hydrostatic Pressure on a Curved Surface

1. Sep 16, 2016

jdawg

1. The problem statement, all variables and given/known data
Gate AB is a quarter-circle 10 ft wide and hinged at B. Find the force F just sufficient to keep the gate from opening. The gate is uniform and weighs 3000 lbf.

Figure and solution attached.
2. Relevant equations

3. The attempt at a solution
I've been able to figure out how to calculate FH and FV. What I'm struggling with is how to find the distances to calculate the moment arms. Its probably something really simple, but I don't understand where they got the 4R/3π or the 2R/π.

I also don't understand how they took two different moments about point B? I only would've thought to do the second moment equation. What do they mean by sum of moments about B of FV?

ΣMB(of FV ) = 8570x = 39936(4.0) - 31366(4.605)
therefore x = 1.787 ft

∑M B(clockwise) = 0 = F(8.0) + (3000)(2.907) - (8570)(1.787) - (19968)(2.667)

Any help is much appreciated!!

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2. Sep 18, 2016

rude man

For the uninitiated, 1 lbf = 4.4482216152605 N.
Is using this dimensional exotica a result of Brexit?

I looked at their derivation and don't fathom (pun?) their rationale.
I would proceed as follows:

at every level of water there is a differential force dF which is ρgx in magnitude and normal to the surface everywhere. x is the distance below the surface (at surface, x = 0).

Find the magnitude of the moment M about B for each level. This is |M| = |r x dF| where r is the vector from B to the level at x. So r and dF are both functions of x.

Then, sum the moment magnitudes and equate to the moment magnitudes formed by the weight of the gate plus the force F.

I hope you have 1st yr calculus. If you haven't had vectors, then |M| = |r|⋅|dF| sinθ where θ is the angle between r and dF. θ is of course also a function of x.

I have a horrible feeling you're supposed to do the problem with what they call FH and FV so I apologize if the above is not much use to you.

EDIT: I changed "torque" to "moment" since that is what they use. Perfectly OK.
2nd EDIT: hint: if F is a normalized vector a i + b j tangent to y = f(x), then the normalized normal is b i +/- a j or a j +/- b i. Pick the appropriate one of the four for your purpose.