Forces on Submerged surfaces—plane surfaces Problem consider a plane surface of area A Draw an y and x axis passing through the centroid x y Place surface in a tank of water with x-axis alighted parallel to the water surface (into page) and y-axis slanted to make an angle of  deg with the water surface  F What is the magnitude F of the resulting Pressure force acting on the surface Of the plane? Where does this Force act on the plane ? The pressure force on the top face is a vector F = (F H, F V ) of magnitude F and direction Inward and normal to the surface. H V

 F The pressure force on the top face is a vector F = (F H, F V ) of magnitude F and direction Inward and normal to the surface. What is the magnitude F of the resulting Pressure force acting on the surface Of the plane? The first question H V The magnitude of the vertical component is The vertical magnitude can also be Equated to the total weight resting above The surface, i.e., Base Vertical Projection of Plane area— Base = A cos  Depth of centroid (1) (2) Equating (1) and (2) Pressure at centroid

Warm ups for the second question Where does this Force act on the plane  F Why is the “center of pressure” cp The location of line of action of the force Below the centroid ? A uniform ally loaded unit beam can be balanced at its centroid But if the force increase down beam Fulcrum needs to be shifted y cp Turning Moments about fulcrum Must balance, i.e., This is exactly The case on a submerged plane The pressure force decrease as we Move toward water surface A single restoring force Of mag. F Under the plate needs To be applied BELOW centroid

Warm ups for the second question A B C Which of these identical plane surface has the largest displacement In the slant (y-direction) between the centroid and center of pressure Displacement increases with slant angle The change in force (change in depth) is larger with a larger slant —on a horizontal plate Displacement decreases with depth—on a very very deep plate Difference in depth along the plate (regardless of its slant) is small And

A single restoring force Of mag. F Under the plate needs To be applied BELOW centroid  F 0 What is the line of action of the resulting Force due to the pressure on the plane Along what line to I have to apply a Single force with magnitude = to The resulting pressure force to Balance the resulting pressure force Do a moment balance about O See book Area moment of inertia about horizontal centroid axis—Page A5 Study example 3.12

In Summary Slant distance to centroid Two forms 3.62

Forces on Submerged surfaces—curved surfaces In Vertical Direction X-distance to centroid –see page A5

Buoyancy Consider a hollow cylindrical steel Buoy cross-section A=5 m 2, volume 10 m 3 Height 2m, Mass 8,000 kg floating at the water air surface What length h of buoy is below the water line ? W FBFB Weight is balanced by upward buoyant force Force balance h Note for this problem if we used a liquid with a specific density < 0.8 The buoy would sink In this case we still have an upward pointing buoyant force BUT we can not displace Enough volume of the liquid to balance the downward pointing weight force.

Net liquid hA

An example Problem Solution Given in class specific gravity < 0.8