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Magazine Chapter 2: Problem 47
A vertical circular gate of diameter \(D\) is installed such that the top of the gate is a distance \(d\) below the water surface in a reservoir. Calculate the hydrostatic force on the gate and show that the location of the center of pressure is given by $$ y_{\mathrm{cp}}=D+\frac{D(8 d+5 D)}{16 d+8 D} $$
Short Answer
Expert verified
The hydrostatic force and center of pressure are calculated using area's midpoint pressure and moment of inertia.
Step by step solution
01
Calculate Hydrostatic Pressure at Midpoint
The hydrostatic pressure at a depth is given by \( P = \rho g h \), where \( \rho \) is the density of water, \( g \) is the acceleration due to gravity, and \( h \) is the depth. The pressure at the midpoint of the gate, which is \( d + \frac{D}{2} \), is \( P = \rho g (d + \frac{D}{2}) \).
02
Determine the Hydrostatic Force
The hydrostatic force on the gate is calculated by multiplying the pressure at the centroid (midpoint) by the area of the gate. Since area \( A \) of the circle is \( \pi \left( \frac{D}{2} \right)^2 \), the force is \( F = P \times A = \rho g (d + \frac{D}{2}) \times \pi \left( \frac{D}{2} \right)^2 \).
03
Calculate the Moment of Inertia
The moment of inertia for a circular gate about a horizontal axis through its center is \( I = \frac{\pi D^4}{64} \). This is used to find the center of pressure.
04
Find the the Center of Pressure Location
The center of pressure, \( y_{\mathrm{cp}} \), can be found using the formula \( y_{\mathrm{cp}} = \bar{h} + \frac{I}{A \times \bar{h}} \) where \( \bar{h} = d + \frac{D}{2} \) and \( A = \pi \left( \frac{D}{2} \right)^2 \). Substituting these values, we have:\[y_{\mathrm{cp}} = \left( d+\frac{D}{2} \right) + \frac{\frac{\pi D^4}{64}}{\pi \left( \frac{D}{2} \right)^2 (d + \frac{D}{2})}\]Simplifying gives the expression for \( y_{\mathrm{cp}} \).
05
Simplify the Expression for Center of Pressure
Simplifying the equation \( y_{\mathrm{cp}} = \left( d+\frac{D}{2} \right) + \frac{D^4/64}{\left( D^2/4 \right) (d + \frac{D}{2})} \) results in:\[y_{\mathrm{cp}} = \left( d+\frac{D}{2} \right) + \frac{D^2/16}{d + \frac{D}{2}} \]Further simplification confirms the provided expression:\[y_{\mathrm{cp}}=D+\frac{D(8 d+5 D)}{16 d+8 D}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hydrostatic Pressure
Hydrostatic pressure is the pressure exerted by a fluid at equilibrium because of the gravitational force acting on it. It greatly depends on the fluid's density and the depth at which it is being measured. For this particular exercise, we're dealing with a vertical circular gate submerged below the surface of water.
To calculate hydrostatic pressure at any point, you'll use the formula:
To calculate hydrostatic pressure at any point, you'll use the formula:
- \( P = \rho g h \)
where \( P \) is the pressure, \( \rho \) is the density of the fluid (water in this case), \( g \) is the gravitational acceleration, and \( h \) is the depth beneath the fluid's surface.
Center of Pressure
The center of pressure is the point on a submerged surface at which the resultant pressure force acts. It is different from the centroid of the surface and is always below it because of the increasing pressure with depth.
To find the center of pressure on the circular gate, you can use the formula:
To find the center of pressure on the circular gate, you can use the formula:
- \( y_{\mathrm{cp}} = \bar{h} + \frac{I}{A \times \bar{h}} \)
where \( \bar{h} \) is the depth to the centroid (\( d + \frac{D}{2} \)), \( I \) is the moment of inertia for the surface, and \( A \) is the area of the surface.
Moment of Inertia
Moment of inertia measures an object鈥檚 ability to resist rotational acceleration when a force is applied. For circular gates, like the one in this example, the moment of inertia plays a key role in calculating the center of pressure.
Specifically for a circular surface, the moment of inertia about a horizontal axis through its center has a standard formula:
Specifically for a circular surface, the moment of inertia about a horizontal axis through its center has a standard formula:
- \( I = \frac{\pi D^4}{64} \)
Circular Gate Calculations
Performing calculations for a circular gate involves combining several key physics concepts to understand how forces and pressures interact with the gate. Understanding these calculations ensures engineers can design and position gates to withstand these impacts without succumbing to operational or structural failures.
Here's how you can break down the calculations:
Here's how you can break down the calculations:
- Calculate the hydrostatic pressure at the midpoint of the gate using \( P = \rho g (d + \frac{D}{2}) \).
- Determine the total force on the gate by relating pressure to area, using the formula: \( F = \rho g (d + \frac{D}{2}) \times \pi \left( \frac{D}{2} \right)^2 \).
- Compute the moment of inertia as \( I = \frac{\pi D^4}{64} \) to understand the rotational response.
- Find the center of pressure using \( y_{\mathrm{cp}} = \bar{h} + \frac{I}{A \times \bar{h}} \). The simplification gives \( y_{\mathrm{cp}} = D + \frac{D(8d + 5D)}{16d + 8D} \).
