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In summary, the conversation discusses a problem involving finding the force F needed to keep a quarter-circle gate with a weight of 3000 lbf from opening. The solution involves calculating the moment arms and summing the moment magnitudes of the water pressure and the weight of the gate and F. The conversation also mentions using vectors and calculus to solve the problem.

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jdawg

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## 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!

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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 d**F** 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 d**F**| where **r** is the vector from B to the level at x. So **r** and d**F** 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**|⋅|d**F**| sinθ where θ is the angle between **r **and d**F.** 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.

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## FAQ: Hydrostatic Pressure on a Curved Surface

## What is hydrostatic pressure?

Hydrostatic pressure is the pressure that is exerted by a fluid at rest. It is caused by the weight of the fluid itself and is perpendicular to the surface of any object immersed in the fluid.

## How does hydrostatic pressure change on a curved surface?

On a curved surface, the hydrostatic pressure is not constant and varies based on the curvature of the surface. The pressure is higher on a concave surface and lower on a convex surface.

## What is the equation for calculating hydrostatic pressure on a curved surface?

The equation for calculating hydrostatic pressure on a curved surface is P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth of the fluid at a specific point on the surface.

## How does the shape of a curved surface affect hydrostatic pressure?

The shape of a curved surface has a significant impact on the hydrostatic pressure. A larger curvature results in a higher pressure, while a smaller curvature leads to a lower pressure. This is due to the increased or decreased area that the fluid is exerting force on.

## What are some practical applications of understanding hydrostatic pressure on a curved surface?

Understanding hydrostatic pressure on a curved surface is crucial in various fields, such as engineering, architecture, and oceanography. It is used to design and construct sturdy structures, calculate the stability of ships and submarines, and study the behavior of fluids in curved containers.

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