From Prob. 3-80, knowing that the density of the suspension depends on liquid depth and changes linearly from 800 kg m to 900 kg/m3 in the vertical direction, determine the resultant force acting on the gate ABC, and its line of action.
Resultant force acting on gate and its line of action.
Answer to Problem 81P
Resultant force acting perpendicular to the gate is
Explanation of Solution
Given:
Density of liquid varies from
Draw the diagram for the cross-section of the gate.
Figure (1)
Draw the diagram for the variation of density with variation of depth.
Figure (2)
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 density as a function of depth.
Here, density is
The density at a particular height is
Write the expression for the area of the elemental strip.
Here, area is
Substitute
Write the expression for the resultant force.
Here, the resultant force 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
Calculation:
Substitute
Substitute
Substitute Equation (VI) in above Equation.
Integrate both sides of the above Equation.
Substitute
Conclusion:
Resultant force acting on gate is
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