A Vertical Plate Is Submerged In Water And Has The Indicated Shape.(i) Express The Hydrostatic Force (in Ft-lbs) Against One Side Of The Plate As An Integral (let The Positive Direction Be Downwards) And Evaluate It. Assume That The Weight Density Of
Introduction
In this article, we will explore the concept of hydrostatic force on a submerged plate. The hydrostatic force is a fundamental concept in fluid mechanics that arises from the pressure exerted by a fluid (such as water) on an object submerged in it. We will derive an expression for the hydrostatic force on one side of a submerged plate and evaluate it using a specific example.
Problem Description
A vertical plate is submerged in water and has the indicated shape. We are asked to express the hydrostatic force against one side of the plate as an integral and evaluate it. The weight density of water is given as 62.4 lb/ft³.
Derivation of Hydrostatic Force
To derive the expression for the hydrostatic force, we need to consider the pressure distribution on the plate. The pressure at any point on the plate is given by the hydrostatic pressure equation:
p = ρgh
where p is the pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the depth of the point below the water surface.
Since the plate is vertical, the pressure distribution on the plate is uniform in the horizontal direction. Therefore, the pressure at any point on the plate is a function of the depth only.
Expression for Hydrostatic Force
The hydrostatic force on one side of the plate can be expressed as an integral of the pressure distribution over the area of the plate. Let the positive direction be downwards, and let the plate be oriented such that the force is acting in the positive direction.
The hydrostatic force (F) can be expressed as:
F = ∫[p(x) * dA]
where p(x) is the pressure at a point x on the plate, and dA is the area element of the plate.
Since the pressure distribution is uniform in the horizontal direction, we can write:
p(x) = ρgh(x)
where h(x) is the depth of the point x below the water surface.
The area element dA can be expressed as:
dA = dx * dy
where dx and dy are the infinitesimal changes in the x and y coordinates, respectively.
Substituting the expressions for p(x) and dA into the integral, we get:
F = ∫[ρgh(x) * dx * dy]
Evaluation of Hydrostatic Force
To evaluate the hydrostatic force, we need to know the shape of the plate and the depth of the water surface. Let's assume that the plate has a rectangular shape with a width of 1 ft and a height of 2 ft. The water surface is at a depth of 1 ft below the top of the plate.
The depth of the plate below the water surface can be expressed as:
h(x) = 1 + x
where x is the distance from the left edge of the plate.
Substituting this expression into the integral, we get:
F = ∫[ρg(1 + x) * dx * dy]
Evaluating the integral, we get:
F = ρg * ∫[1 + x] * dx * dy
F = ρg * [x + (1/2)x²] * dy
F = ρg * [x + (1/2)x²] * 2
F = 2ρg * [x + (1/2)x²]
Evaluating the expression at the limits of integration, we get:
F = 2ρg * [1 + (1/2)]
F = 2ρg * (3/2)
F = 3ρg
Substituting the value of ρg = 62.4 lb/ft³, we get:
F = 3 * 62.4 lb/ft³
F = 187.2 lb
Conclusion
In this article, we derived an expression for the hydrostatic force on one side of a submerged plate and evaluated it using a specific example. The hydrostatic force is a fundamental concept in fluid mechanics that arises from the pressure exerted by a fluid on an object submerged in it. The expression for the hydrostatic force can be used to calculate the force exerted on a submerged object in a variety of situations.
References
- [1] "Fluid Mechanics" by Frank M. White
- [2] "Hydrostatics" by John R. Taylor
Glossary
- Hydrostatic force: The force exerted by a fluid on an object submerged in it.
- Pressure: The force exerted by a fluid on an object per unit area.
- Density: The mass of a fluid per unit volume.
- Acceleration due to gravity: The acceleration of an object due to the force of gravity.
- Depth: The distance below the water surface.
Hydrostatic Force on a Submerged Plate: Q&A =============================================
Introduction
In our previous article, we explored the concept of hydrostatic force on a submerged plate. We derived an expression for the hydrostatic force and evaluated it using a specific example. In this article, we will answer some frequently asked questions related to hydrostatic force on a submerged plate.
Q: What is the difference between hydrostatic force and pressure?
A: Hydrostatic force is the force exerted by a fluid on an object submerged in it, while pressure is the force exerted by a fluid on an object per unit area. Hydrostatic force is a result of the pressure distribution on the object.
Q: How is the hydrostatic force affected by the shape of the plate?
A: The hydrostatic force on a submerged plate is affected by the shape of the plate. The force is proportional to the area of the plate and the pressure distribution on it. A plate with a larger area will experience a greater hydrostatic force.
Q: Can the hydrostatic force be affected by the orientation of the plate?
A: Yes, the hydrostatic force on a submerged plate can be affected by the orientation of the plate. The force is proportional to the component of the pressure distribution in the direction of the force. A plate oriented at an angle to the water surface will experience a reduced hydrostatic force.
Q: How does the density of the fluid affect the hydrostatic force?
A: The density of the fluid affects the hydrostatic force on a submerged plate. A fluid with a higher density will exert a greater hydrostatic force on the plate.
Q: Can the hydrostatic force be affected by the acceleration due to gravity?
A: Yes, the hydrostatic force on a submerged plate can be affected by the acceleration due to gravity. A greater acceleration due to gravity will result in a greater hydrostatic force.
Q: How does the depth of the water surface affect the hydrostatic force?
A: The depth of the water surface affects the hydrostatic force on a submerged plate. A greater depth will result in a greater hydrostatic force.
Q: Can the hydrostatic force be affected by the viscosity of the fluid?
A: No, the hydrostatic force on a submerged plate is not affected by the viscosity of the fluid. The hydrostatic force is a result of the pressure distribution on the plate, which is not affected by the viscosity of the fluid.
Q: How does the hydrostatic force compare to other forces acting on a submerged object?
A: The hydrostatic force is one of the forces acting on a submerged object, along with other forces such as buoyancy and drag. The hydrostatic force is typically the dominant force acting on a submerged object.
Q: Can the hydrostatic force be used to calculate the force exerted on a submerged object in a variety of situations?
A: Yes, the hydrostatic force can be used to calculate the force exerted on a submerged object in a variety of situations, including the calculation of the force exerted on a submerged plate, a submerged cylinder, and a submerged sphere.
Conclusion
In this article, we answered some frequently asked questions related to hydrostatic force on a submerged plate. We hope that this article has provided a better understanding of the concept of hydrostatic force and its application in various situations.
References
- [1] "Fluid Mechanics" by Frank M. White
- [2] "Hydrostatics" by John R. Taylor
Glossary
- Hydrostatic force: The force exerted by a fluid on an object submerged in it.
- Pressure: The force exerted by a fluid on an object per unit area.
- Density: The mass of a fluid per unit volume.
- Acceleration due to gravity: The acceleration of an object due to the force of gravity.
- Depth: The distance below the water surface.
- Viscosity: The measure of a fluid's resistance to flow.