The gate OAB under water is 5 m-wide and is hinged along O as shown.
(a) Find the hydrostatic force on the horizontal surface AB.
(b) Determine the magnitude and point of action of the hydrostatic resultant force on the vertical surface OA.
(c) Draw a free body diagram of the gate and find the horizontal force at point A required to support the gate from opening. Neglect the weight of the gate.
To find the respective hydrostatic forces on the gate:
(a) The hydrostatic force on the horizontal surface AB is given by the formula:
F = ρ * g * A
where ρ is the density of the fluid, g is the acceleration due to gravity, and A is the area of the surface in contact with the fluid.
In this case, the surface AB is horizontally submerged, so the hydrostatic force can be found as follows:
F = ρ * g * A
= ρ * g * (width) * (length)
= ρ * g * 5 * (length)
where the width of surface AB is given as 5 meters.
(b) The magnitude of the hydrostatic resultant force on the vertical surface OA is found using the formula:
F = ρ * g * A
In this case, the surface OA is a vertical surface, so the hydrostatic force can be calculated as follows:
F = ρ * g * A
= ρ * g * (width) * (depth)
= ρ * g * 5 * (depth)
where the width of surface OA is 5 meters.
To find the point of action for this force, we need to determine the centroid of the vertical surface OA. The centroid is given by the formula:
x̄ = (2/3) * (depth)
This means that the horizontal distance of the point of action from O is (2/3) times the depth of the surface OA.
(c) To find the horizontal force (at point A) required to support the gate from opening, we need to consider the moment equilibrium of the gate about point O.
Since the gate is hinged at point O, only the horizontal components of the hydrostatic forces on surfaces AB and OA will contribute to the moment equation. The moment equation can be written as:
ΣM = 0
= (Hydrostatic force on AB) * (horizontal distance from O to AB)
+ (Hydrostatic force on OA) * (horizontal distance from O to OA)
- (Horizontal force at point A) * (distance from O to A)
Simplifying the equation, we have:
(Hydrostatic force on AB) * (horizontal distance from O to AB)
+ (Hydrostatic force on OA) * (horizontal distance from O to OA)
= (Horizontal force at point A) * (distance from O to A)
Substituting the values obtained in parts (a) and (b), we can solve for the horizontal force at point A.
To find the answers, we will be using the principles of hydrostatics, which is the study of fluids at rest. Here's how we can solve each part of the problem:
(a) To find the hydrostatic force on the horizontal surface AB, we need to calculate the pressure at different points along that surface and then integrate it over the surface area.
1. First, we need to determine the depth of each point on the surface AB. Since the gate is submerged under water, the depth varies along the surface AB. Let's call the depth at a particular point x.
2. The pressure at a specific depth in a fluid is given by the equation p = ρgh, where p is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth.
3. Since the gate is submerged under water, the fluid is water and its density (ρ) is approximately 1000 kg/m^3.
4. The depth at any point x on the horizontal surface AB can be determined by subtracting x from the distance from point A to point O (OA). Let's call this distance y.
5. Therefore, the pressure at any point (x) on the surface AB is given by p = 1000 * 9.8 * (y - x).
6. Now, we integrate this pressure equation over the surface AB, with x ranging from 0 to 5 meters (the width of the gate):
Force on AB = ∫(0 to 5) (1000 * 9.8 * (y - x)) dx
7. Integrating the equation, we get:
Force on AB = 1000 * 9.8 * [y * x - (x^2 / 2)] evaluated from x = 0 to x = 5
By plugging in the value of y (the distance from point A to point O), you can calculate the hydrostatic force on the horizontal surface AB.
(b) To determine the magnitude and point of action of the hydrostatic resultant force on the vertical surface OA, we can use the concept of centroid.
1. The centroid of a submerged vertical surface is located at a distance of 2/3 of the height from the bottom of the surface.
2. In this case, the height of the gate is the same as the width (5m). So, the centroid will be at a distance of 2/3 * 5 = 10/3 meters from the bottom of the surface.
3. The magnitude of the hydrostatic resultant force is calculated using the equation:
Resultant force = pressure at centroid * area of the surface.
4. The pressure at the centroid (p) is given by p = 1000 * 9.8 * (10/3).
5. The area of the vertical surface OA is 5 * 5 = 25 square meters.
By plugging in the values, you can calculate the magnitude of the hydrostatic resultant force and the point of action on the vertical surface OA.
(c) To find the horizontal force at point A required to support the gate from opening, we need to consider the equilibrium of forces acting on the gate.
1. Draw a free body diagram of the gate, labeling all the forces acting on it. These forces include the hydrostatic force on surface AB (found in part (a)), the horizontal force at point A, and any other external forces.
2. Since the gate is in equilibrium, the sum of all the forces in the horizontal direction should be zero.
3. Write an equation by summing up all the horizontal forces and setting them equal to zero.
By solving this equation, you can find the horizontal force required to support the gate from opening at point A.