Hydrostatic Pressure on a Curved Surface

[jdawg]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I've been able to figure out how to calculate F_H and F_V. 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 F_V?

ΣM_B(of F_V ) = 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!

 

[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 F_H and F_V 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.