This Demonstration determines the magnitude of the applied force needed to keep a gate, which is
meters wide and
meters tall, closed (see Figure 1).
The magnitude of the resultant force is found by summing the differential forces over the gate surface:
is the vertical distance from the water surface (m),
coordinate (along the diagonal) from the water surface (m),
is the resultant force (N),
is the gate area (
is the width of the gate (m),
is the length of the gate (m) as shown in Figure 1,
is the angle (degrees) and
is the specific weight of water (
Solving the integral for
is the vertical distance from the fluid surface to the centroid of the gate (m) and
coordinate of the gate centroid (m).
of the resultant force can be found by summing moments around the hinge:
is substituted into this equation, and the right side is integrated from
A moment balance is done to determine the applied force that keeps the gate closed:
rearranging to solve for
is the weight of the gate (N),
is the diagonal length from the hinge to the top of the water (m) and
is the applied force (N).
 B. R. Munson, D. F. Young, T. H. Okiishi and W. W. Huebsch, Fundamentals of Fluid Mechanics
, 6th ed., Hoboken, NJ: John Wiley and Sons, 2009 pp. 58–60.