Archimedes Principle

Archimedes (287-212 B.C.)

Any body immersed in a fluid is subjected to an upward force called buoyant force equal to the weight of the displaced fluid.
 

Where
$BF$ = buoyant force
$\gamma$ = unit weight of fluid
$V_D$ = volume of fluid displaced by the body
 

Situation
The figure below shows a vertical circular gate in a 3-m diameter tunnel with water on one side and air on the other side.
 

1. Find the horizontal reaction at the hinge.
   A.   412 kN
   B.   408 kN
   C.   410 kN
   D.   414 kN
2. How far from the invert of the tunnel is the hydrostatic force acting on the gate?
   A.   1.45 m
   B.   1.43 m
   C.   1.47 m
   D.   1.41 m
3. Where will the hinge support be located (measured from the invert) to hold the gate in position?
   A.   1.42 m
   B.   1.46 m
   C.   1.44 m
   D.   1.40 m

Total Hydrostatic Force on Plane Surfaces
For horizontal plane surface submerged in liquid, or plane surface inside a gas chamber, or any plane surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by
 

where p is the uniform pressure and A is the area.
 

In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its center of gravity.
 

where p_cg is the pressure at the center of gravity. For homogeneous free liquid at rest, the equation can be expressed in terms of unit weight γ of the liquid.
 

where   $\bar{h}$   is the depth of liquid above the centroid of the submerged area.
 

Problem
How high will liquid rise in the piezometers shown in the figure if the pressure at M is 68.95 kPa and the liquid is (a) water, (b) oil (sp gr 0.85), (c) mercury, (d) brine (sp gr 1.15)?
 

Problem
In the piezometers of the figure shown, liquid stands 1.37 m above point M. What is the pressure at M in kiloPascal if the liquid is (a) water, (b) oil (sp gr 0.90), (c) mercury, and (d) molasses (sp gr 1.5).
 

Manometer is a simple and inexpensive device of measuring pressure and pressure difference. It is usually bent to form a U-tube and filled with liquid of known specific gravity. The surface of the liquid will move in proportion to changes of pressure.
 

Problem
In a condenser containing air and water, the air pressure is 22.1 kPa absolute. What is the gage pressure in kiloNewton per square meter at a point 1.37 m below the water surface.
 

Problem
A pressure gage on the discharge side of a pump reads 300 kPa. Oil (sp gr 0.82) is being pumped. Compute the pressure head in meters of oil.
 

Problem
A gage on the suction side of a pump shown vacuum of 254 mm of mercury. Compute (a) pressure head in meters of water; (b) pressure in kiloPascal; (c) absolute pressure in kiloPascal if barometer reads 736.6 mm of mercury.
 

Problem
The pressure in a gas tank is 2.75 atmospheres. Compute the pressure in kiloPascal and the pressure head in meter of water.