Problem
The 200-kg, 5-m-wide rectangular gate shown in Fig. P10-39 is hinged at B and leans against the floor at A making an angle of 45° with the horizontal. The gate is to be opened from its lower edge by applying a normal force at its center. Determine the minimum force F required to open the water gate.
Step-by-step solution
Step 1 of 8
Draw the free body diagram of the gate.
Here,
is the hydrostatic force on the gate due to the fluid lying above, C is the centroid of the gate, 
is a point such that it is along the inclination of gate and lies at the free surface, 
is the distance of the centroid of the gate from the free surface, 
is the distance of the centroid of the gate along the inclination of gate, 
is the angle of inclination of gate AB with the horizontal, 
is the position of center of pressure, and 
is the weight of the gate.Comment Step 2 of 8
Find the length of the gate AB from principles of trigonometry.
Calculate the length BC.
Substitute
for 
.
Comment Step 3 of 8
Calculate the distance between the free surface and the centroid of the gate.
Substitute
for BC.
Calculate the area of the gate.
Here, w is the width of the gate.Substitute
for 
and 
for 
.
Comment Step 4 of 8
Write the expression for hydrostatic force on the gate.
Substitute
for 
, 
for 
, 
for 
, and 
for 
.
Comment Step 5 of 8
Calculate the moment of inertia of the gate about the
axis.
Substitute
for 
and 
for 
.
Write the expression for depth of center of pressure from the free surface.
Substitute
for 
, 
for 
, 
for 
, and 
for
.
Calculate the distance of the pressure center from the point D from the concept of trigonometry.
Substitute
for 
and 
for 
.
Comment Step 6 of 8
Calculate the moment about B applied by the weight of the gate and the resultant hydrostatic force on the gate.
Calculate the moment about B due to the applied force F.
Consider the requirement of the question and understand that to lift the gate the moment applied by force F must be equal to that B applied by the weight of the gate and the resultant hydrostatic force on the gate.
Substitute
for 
and 
for 
.
Substitute
for 
, 
for 
, 
for 
, 
for 
, and 
for 
.
Therefore, the minimum force required to open the water gate is
.Comment Step 7 of 8
The given problem can be solved using EES software by using the following steps.1. Open EES software2. Create a new file by clicking on the ‘New’ of the ‘File menu’Now type the following code in dailog box.
Comment Step 8 of 8
Click on solve button or press “F2” key on the keyboard. The solution window pops up.
From the results the F value is,
Therefore, the minimum force required to open the gate is
Comment
