Question

2) A thin 4-ft-wide, right-angle gate with negligible mass is free to pivot about a frictionless hinge at point O. The horizontal portion of the gate covers a 1-ft-diameter drainpipe which contains air at atmospheric pressure. Determine the minimum water depth h at which the gate will pivot to allow water to flow through the pipe.
Ans: h = 1.88 ft

Answer 1

Concepts and reason

Hydrostatic force:

It refers to the force generated by the action of liquid’s on the surface of immersed plane

Specific weight:

It refers to the weight per unit volume of the material. It is denoted by and its unit is.

Moment of a force:

Moment of a force refers to the propensity of the force to cause rotation on the body it acts upon. The moment’s magnitude can be determined from the product of force’s magnitude and the distance to the force measured perpendicularly.

Determine the minimum water depth by applying equilibrium condition for moments about the point O.

Fundamentals

Write the expression for calculating moment about point.

Here, magnitude of the force is and its perpendicular distance is .

Write the equilibrium condition for moment about any point.

Here, sum of all moments about the point is .

The formula to calculate the magnitude of the resultant force in terms of pressure is as follows:

Here, uniform pressure on the bottom is p and area of the bottom is A.

The formula to calculate the pressure for an open tank is as follows:

$p=yh $

Here, specific weight of the fluid in the tank is and depth of the fluid is h.

The formula to calculate the hydrostatic force acting on a fluid is as follows:

Here, vertical distance from the fluid surface to the centroid of the area is and centroid of the area is A.

Show the free-body diagram of the right angled gate as in Figure (1).

Here, magnitude of the resultant fluid force on the vertical and horizontal portion of the gate is respectively, length from the hinge O to the radius of the drain pipe is , length from the hinge vertically to the centroid of the right angle gate is , horizontal and vertical portion reaction forces are respectively.

Calculate the resultant fluid force on the vertical portion of the gate.

Here, area of rectangular surface on the vertical portion of gate is , breath and height on the vertical portion of gate is , weight density of gate is and vertical distance from the fluid surface to the centroid of the area on the vertical portion of gate is .

The average pressure occurs at a depth . Thus, the fluid force on the vertical portion of the gate is,

$(4x9)x=* $

Substitute for and for .

Calculate the resultant fluid force on the horizontal portion of the gate.

Here, area of the horizontal portion of the plate is , diameter of pipe on the horizontal portion of gate is and vertical distance from the fluid surface to the centroid of the area on the horizontal portion of gate is .

Substitute for and for .

Apply equilibrium condition for moments about point O.

Substitute for , for , 3 ft for , and for.

Ans:

The minimum depth of water is .