Consider a 1-m wide inclined gate of negligible weight that separates water from another fluid. What would be the volume of the concrete block
FIGURE P3−92
The volume of the concrete block.
Answer to Problem 92P
The volume of the concrete block is
Explanation of Solution
Given information:
Specific gravity of the concrete is
The below figure shows the free body diagram of the gate.
Figure-(1)
Write the expression for the length of the portion BC.
Here, the angle from negative x axis is
Write the expression for the length of the height DC.
Write the expression for the vertical depth of the centre of gravity of the gate in contact with water from the free surface.
Here, the vertical height of the gate in contact with water is
Write the expression for the pressure of water acting on the gate.
Here, the acceleration due to gravity is
Write the expression for the resultant force acting on the triangular gate.
Here, the area is
Write the expression for the density of tetrachloride is
Write the expression for the vertical depth of the centre of gravity of the gate in contact with the carbon tetrachloride from the free surface.
Here, the height of the gate in contact with the carbon tetrachloride from the free surface is
Write the expression for the resultant force of carbon tetrachloride acting on the gate.
Here, the area of resisting pressure of carbon tetrachloride is
Write the expression for the pressure of the carbon tetrachloride acting on the gate.
Here, the density of carbon tetrachloride is
Write the expression for the area of the resisting pressure of carbon tetrachloride.
Here, the width of the gate is
Write the expression for the point of the application of force due to the water.
Here, the moment of inertia about x axis and point C is
Write the expression for the moment of inertia about x axis and for portion BC.
Substitute
Write the expression for the point of the application of force due to the carbon tetrachloride.
Write the expression for the moment of inertia about x axis and for portion DC.
Substitute
Write the expression for the moments about point C.
Here, the tension in the string is
Below figure shows the free body diagram of the block.
Figure-(2)
Write the expression for the force balance in vertical direction.
Here, the weight of the concrete block is
Write the expression for the weight of the block.
Here the volume of the block is
Write the expression for the density of the concrete.
Here, the specific gravity of the concrete block is
Substitute
Write the expression for the buoyancy force.
Here, the displaced volume is
Write the expression for the area of the resisting pressure of water.
Write the expression for the density of the carbon tetrachloride.
Here, the specific gravity of the carbon tetrachloride is
Calculation:
Substitute
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Conclusion:
The volume of the concrete block is
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Chapter 3 Solutions
FLUID MECHANICS:FUND.+APPL.(LL)>CUSTOM<
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