Question

   
A homogeneous, 4 ft wide, 8 ft long rectangle gate weighing 800 lbis held in place by a horizontal flexible cable. Water acts againstthe gate, which is hinged at point A. friction in the hinge isnegligible. Determine the tension in the cable.(answer=1350 lb)

Answer 1

Concepts and reason

Pascal’s law states that at any point in a fluid, pressure is independent of motion and direction if there is no shear stress.

Consider the hydrostatic forces on an inclined plate submerged in a fluid.

Resultant force on the plate is given as follows:

$Fr=yhA $

Here, specific weight of the fluid is , location of centroid of the plate from the free surface of water is h, and area of the plate is A.

The resultant force acts perpendicular to the gate.

Write the point of action of the resultant force on the plate.

$+a=* $

Here, distance of the centroid along the inclined plate from the free surface is and moment of inertia of the plate along centorial axis is .

Write the point of action of the resultant force on the plate along the depth.

Fundamentals

Moment of inertia of rectangular plate about its centroid is given by

Here, base length of the rectangle is b and height of the rectangle is h.

Specific weight ():

It is the weight of the fluid per unit volume and it is equal to product of density and acceleration due to gravity.

Specific weight of water is .

Draw free body diagram of the plate.

Calculate the center of the plate from free surface of water, h.

Determine the center of pressure of the point from the free surface of water.

Substitute for b, for d, for h, and for .

Calculate the hydrostatic force on the plate.

Substitute for h, for , for b, and for d.

Consider moments about point A in the free body diagram.

Substitute for W, for , and for .

Ans:

The tension in the cable is .