Problem 25 - Calculus of a Single Variable

Question

A rectangular plate of height \(h\) feet and base \(b\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center is \(k\) feet below the surface of the fluid, where \(k>h / 2\). Show that the fluid force on the surface of the plate is \(F=w k h b\).

Step-by-Step Solution

Answer

The fluid force on a submerged rectangular plate of dimensions \(h\) feet by \(b\) feet, with its center \(k\) feet below the surface of a fluid of weight density \(w\) pounds/cubic feet, is calculated as \(F = wkhb\). The most important part is understanding the pressure due to fluid on small strips of the plate and then summing them up.

Step 1: Understand the problem

A rectangular plate of height \(h\) feet and base \(b\) feet is submerged vertically in a fluid. The center of the plate is \(k\) feet below the surface, where \(k > h/2\). We have to show that the fluid force on the surface of the plate is given by the formula \(F = wkhb\), where \(w\) is the weight of the fluid per cubic foot.

Step 2: Determine pressure on an infinitesimal strip

Let's consider a tiny horizontal strip of the rectangular plate at a depth \(y\) from the surface of the fluid. Because the pressure exerted by a fluid column is given by \(P = w \cdot y\), where \(w\) is the weight density of the fluid and \(y\) is the depth, the fluid pressure on this small strip is \(w \cdot y\). This strip has dimensions \(b\) and \(dy\) (where \(dy\) is the infinitesimal height) and thus, its area is \(b \cdot dy\). So, the force \(dF\) on this strip by the fluid is \(w \cdot y \cdot b \cdot dy\). This is because force is pressure times area.

Step 3: Sum up the forces on all such strips

Now, our plate is submerged vertically with its top \(k - h/2\) feet and bottom \(k + h/2\) feet below fluid surface. Therefore, to calculate total fluid force on the plate, we need to sum the forces on all such strips from the top of the plate to the bottom, i.e., we need to integrate the infinitesimal force \(dF\) from \(y = k - h/2\) to \(y = k + h/2\). So, the total force \(F\) is given by the integral \(\int_{k-h/2}^{k+h/2} w \cdot y \cdot b \, dy\)

Step 4: Evaluate the integral

Calculate the integral \(\int_{k-h/2}^{k+h/2} w \cdot y \cdot b \, dy\). Solving this integral, the fluid force \(F\) on the plate is calculated as \(F = w \cdot b \cdot [(k+h/2)^2/2 - (k-h/2)^2/2]\). Simplifying the above expression, we get \(F = wkhb\). Thus, we have shown that the fluid force is indeed given by \(F = wkhb\), as required.

Key Concepts

Pressure on Submerged Surfaces

When a surface is submerged in a fluid, the pressure exerted on it depends on the depth. This is because the deeper you go, the more fluid weight is pressing down, increasing the pressure. Here's how it works:

• Pressure is defined as force exerted per unit area.
• In fluids, pressure increases with depth. The formula for pressure at a depth is given by: \( P = w \cdot y \), where \(w\) is the weight density of the fluid and \(y\) is the depth.
• This means the farther down you measure, the greater the pressure, as more fluid is above that point.

The calculation of pressure on submerged surfaces becomes important in fields like engineering and fluid mechanics, especially when dealing with structures submerged in water or other fluids, assuring safety and effectiveness.

Integration in Calculus

Integration in calculus is a powerful tool that lets us calculate things like areas and quantities under curves. It's particularly useful in finding total values from rates or densities. Here's why it's crucial:

• Integration sums up smaller parts, a process called 'accumulation'.
• In this exercise, it helps us find the total fluid force on the plate by summing the infinitesimal forces on its small strips.
• The integral \( \int_{k-h/2}^{k+h/2} w \cdot y \cdot b \, dy \) sums up all these small forces from the top to the bottom of the plate.

This approach is vital for calculating fluid forces because it provides a comprehensive sum based on variable pressure over the submerged surface.

Fluid Mechanics

Fluid mechanics is the study of fluids (liquids and gases) and the forces on them. It plays a key role in understanding how fluids behave when they're at rest or in motion. Here’s a simple breakdown:

• Fluids exert pressure on anything submerged in them, from all sides.
• The basic concept of pressure plays a critical role in how submarines, dams, and many engineering projects are designed.
• Fluid mechanics principles help us solve complex problems, like analyzing the force on submerged structures and calculating the effective design requirements.

Understanding fluid mechanics is crucial for developing safe and efficient designs in engineering and related fields.

Rectangular Plate Submerged in Fluid

A rectangular plate brings a straightforward model to study when submerged in fluid. Essential for engineering, the understanding of force distribution on such plates can be simplified:

• The plate's dimensions allow us to calculate the area exposed to fluid pressure.
• Being submerged vertically means different parts of the plate experience varying pressures due to depth differences.
• The force on the plate is calculated using integration, considering varying pressure over its height and constant width.

This analysis helps engineers design structures like underwater gates or panels efficiently, ensuring they can withstand the pressures exerted by liquids.