Question

8. An open tank has a vertical partition and on one side contains gasoline with a density p = 700 kg/m3 at a depth of 4 m, as shown in Figure 8. A rectangular gate that is 4 m high and 2 m wide and hinged at one end is located in the partition. Water is slowly added to the empty side of the tank. At what depth, h, will the gate start to open?

Answer 1

Concepts and reason

Hydrostatic force:

Hydrostatic force refers to the force exerted by fluid on walls of container, dam and ships.

Centroid:

Centroid is the mean position of all points in a geometric figure. It is the point whose coordinates are the mean of all points in the geometric area.

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 magnitude of the moment can be determined from the product of force’s magnitude and the perpendicular distance to the force.

Specific weight:

Specific weight is the ratio of weight to the unit volume. It is denoted by symbol .

Density:

Density refers to the mass occupied by a substance per unit volume. It is denoted by symbol and its unit is.

Firstly, the resultant force acting on the gate by the gasoline is calculated by using the expression of hydrostatic force. Then, the resultant force acting on the gate by the water is calculated by using the expression of hydrostatic force.

Finally, the depth of the water at which the gate will start to open is obtained by using equilibrium condition of moment about the hinge.

Fundamentals

The expression for a hydrostatic force is given as follows:

Here, hydrostatic force is , density is , acceleration due to gravity is , specific weight is , centroid is and area is .

Equilibrium condition for a body:

For a body to be in equilibrium in two dimensional planes, the conditions which are required to be satisfied are,

Write the force equilibrium equation in the x direction.

Here, the total force acting in x direction is .

Write the force equilibrium equation in the y direction.

Here, the total force acting in y direction is .

Write the formula to calculate moment about any point A due to force .

Here, the perpendicular distance of force from the point A is .

The summation of moment acting about any point will be equal to zero.

Draw the free body diagram of gate as shown in Figure (1).

Here, hydrostatic force acting on the gate by the gasoline is , hydrostatic force acting on the gate by the water is , height of the water level is , height of the gasoline level is , vertical reaction at hinge is and horizontal reaction at hinge is .

Write the equation of hydrostatic force acting on the gate by the gasoline.

Here, density of the gasoline is , centroid of the force acting on gate by the gasoline from the surface level is and surface area of the gasoline on gate is .

Substitute for , for g, for and for .

Write the equation of hydrostatic force acting on the gate by the water.

Here, density of the water is , centroid of the force acting on gate by the water from the surface level is and surface area of the water on gate is .

Substitute for , for g, for and for .

Apply equilibrium condition for moment about hinge.

Here, distance measured from the bottom surface to the force acting by water on the gate is and distance measured from the bottom surface to the force acting by gasoline on the gate .

Substitute for , for , for and for .

Ans:

The depth of the water at which the gate will start to open is 3.55 m.