A long, solid cylinder of radius 2 ft hinged at point A is used as an automatic gate, as shown in Fig. P3-79E. When the water level reaches 12 ft, the cylindrical gate opens by turning about the hinge at point A. Determine (a) the hydrostatic force acting on the cylinder and its line of action when the gate opens and (b) the weight of the cylinder per ft length of the cylinder.
(a)
The hydrostatic force when gate opens.
Answer to Problem 79EP
The hydrostatic force when gate opens is
Explanation of Solution
Given:
Water level is
Radius of the cylinder is
Draw a free body diagram acting on the cylinder as shown in the below figure.
Figure (1)
Write the expression for the horizontal force on the vertical surface.
Here, the horizontal force on vertical surface is
Write the expression for the area.
Here, radius is
Write the expression for the distance from surface to the centre of gravity.
Substitute the Equation (II) and Equation (III) in Equation (I).
Calculation:
Substitute
Conclusion:
Hydrostatic force is
(b)
Weight of the cylinder per unit length.
Answer to Problem 79EP
Weight of the cylinder per unit length is
Explanation of Solution
Write the expression for the vertical force on the projected horizontal surface.
Here, height of the liquid column from free surface of water till the bottom of the gate is
The figure below shows the diagram for the volume enclosed per unit width.
Figure-(2)
Write the expression for the volume of element.
Here, radius is
Write the expression for the weight of the element per unit width of the gate.
Here, weight is
Write the expression for the vertical force.
Here, vertical force is
Write the expression for the resultant force.
Here, resultant force is
Write the expression for the direction of hydrostatic force.
The below figure represents direction of hydrostatic force.
Figure-(3)
Write the expression for the moment about the hinged point.
Here, the weight of the cylinder is
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The weight of the cylinder is
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