An open settling tank shown in the figure contains a liquid suspension. Determine the resultant force acting on the gate and its line of action if the liquid density is 850 kg/m3. The gate is parabolic as sketched, looking straight at the gate.
The Resultant force acting on gate and its line of action.
Answer to Problem 80P
The Resultant force acting on gate is
Explanation of Solution
Given:
Density of liquid is
Draw the cross-sectional view of the gate.
Figure (1)
Write the expression for the curve.
Here, horizontal axis is denoted by the x and vertical axis is denoted by the y.
Write the expression for the vertical distance between the centre of the elemental axis and the x axis.
Here, the vertical distance between the centre of the elemental axis and the x axis is
Write the expression for the area of the elemental strip.
Here, area is
Write the expression for the centre of gravity.
Here, centre of gravity is
Substitute
Draw a view of gate and liquid.
Figure (2)
Write the expression for the length of gate which is in contact with the liquid.
Here, length of the gate is
Write the expression for the centre of gravity of the gate from the free surface.
Here, the mass moment of inertia is
Write the expression for the vertical depth of the centre of gravity of gate from the free surface.
Here, the vertical depth of the centre of gravity of gate from the free surface is
Write the expression for pressure acing on the gate.
Here, pressure acing on the gate is
Write the expression for the area of gate resisting the pressure on the gate.
Here, the area of gate resisting the pressure on the gate is
Write the expression for the resultant force acting on the gate.
Here, the resultant force acting on the gate is
Write the expression for the moment of inertia.
Write the expression for the action of resultant hydrostatic force.
Here, the action of resultant hydrostatic force is
Write the expression for the centre of line of action from the free surface.
Here, the centre of line of action from the free surface is
Write the expression for the centre of location of the force from the bottom.
Here, the centre of location of the force from the bottom is
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Draw the diagram for different parameter.
Figure (3)
Substitute
Substitute
Conclusion:
Resultant force acting on gate is
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Chapter 3 Solutions
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- International Edition---engineering Mechanics: St...Mechanical EngineeringISBN:9781305501607Author:Andrew Pytel And Jaan KiusalaasPublisher:CENGAGE L