Problem 67 - Essential Calculus

Question

Fluid Force on a Rectangular Plate 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 \(h \leq k / 2\). Show that the fluid force on the surface of the plate is \(\boldsymbol{F}=w k h b\)

Step-by-Step Solution

Answer

The fluid force on the surface of the plate \(F = w * k * h * b\) pounds.

Step 1: Identify the pressure

Firstly, identify the pressure at the center of the plate. The pressure \(p\) at depth \(k\) in a fluid is given by the equation \(p = w * k\) where \(w\) is the weight per cubic foot of the fluid and \(k\) is the depth from the surface.

Step 2: Calculate the force

Since the pressure is constant over the height of the plate due to its center being more than half the height from the surface, the force \(F\) is given by the product of pressure \(p\) and the area of the plate \(A = h * b\). Hence, the fluid force on the plate is \(F = p * A = w * k * h * b\).

Key Concepts

Pressure in Fluids

Pressure in fluids is a fundamental concept in fluid mechanics that helps us understand how forces are distributed within a fluid. This pressure is defined as the amount of force exerted by the fluid per unit area. In mathematical terms, when we talk about pressure at a specific depth, it is calculated using the formula:

• \[ p = w \times k \]

where \( p \) denotes the pressure, \( w \) is the fluid's weight per cubic foot, and \( k \) is the depth below the fluid surface.

It's essential to grasp that pressure increases with depth, thus the deeper you go, the greater the pressure. This is because the weight of the fluid above contributes to the overall force exerted at deeper levels.

Rectangular Plate Submersion

When a rectangular plate is submerged vertically in a fluid, several factors determine the total fluid force exerted on it. Here, we're particularly focused on the physical dimensions of the plate, which are given by the height \( h \) and base \( b \). The unique feature of submersion is how the depth affects pressure.

For this specific scenario, the center of the plate is at a depth of \( k \) feet, allowing us to consider the entire pressure as acting uniformly across the surface of the plate since \( h \leq k/2 \). This means the top and bottom of the plate are at similar pressures, simplifying the calculations for fluid force.

Fluid Mechanics

Fluid mechanics is the study of fluids and how they behave when at rest or in motion. It examines the principles that govern the behavior of gases and liquids. In this context, understanding how forces act through fluids is central. One key concept is the relationship between pressure and force: force is the pressure exerted over an area.

Taking this into account, the force created from fluid pressure can exert itself over surfaces like submerged plates. By knowing the pressure exerted by the fluid and the surface area of the plate, we can calculate the force using the flow of fluid mechanics principles.

Surface Area Calculation

To find the fluid force acting on a submerged rectangular plate, calculating the surface area is crucial. The plate's surface area is derived from its dimensions: height \( h \) and base \( b \). This calculation is straightforward:

• \[ A = h \times b \]

This formula gives us the total area over which the pressure from the fluid will act.

With the area and pressure known, the fluid force \( F \) can be found using:

• \[ F = p \times A \]

where \( F \) is the fluid force, illustrating how area is a direct multiplier of the pressure exerted by the fluid on the plate.