Problem 45 - Thomas Calculus in SI Units

Question

A vertical rectangular plate \(a\) units long by \(b\) units wide is submerged in a fluid of weight-density \(w\) with its long edges parallel to the fluid's surface. Find the average value of the pressure along the vertical dimension of the plate. Explain your answer.

Step-by-Step Solution

Answer

The average pressure is \(\frac{w a}{2}\).

Step 1: Understanding Pressure in Fluid

The pressure at any depth in a fluid is given by the product of the weight-density of the fluid and the depth of the point. Hence, the pressure at a depth of \(y\) units is \(P(y) = w \, y\).

Step 2: Setup the Expression for Average Pressure

To find the average pressure along the vertical dimension of the plate, you need to calculate the average of the pressure values along the depth \(y\), from the top of the plate (\(y = 0\)) to the bottom (\(y = a\)). Mathematically, the average pressure over a length \(a\) is given by \(\frac{1}{a} \int_0^a P(y) \, dy\).

Step 3: Integrate the Pressure Function

Substitute \(P(y) = w \, y\) into the integral. Thus, the integral becomes \(\int_0^a w \, y \, dy\). This integral evaluates to \(\left[\frac{w}{2} y^2\right]_0^a = \frac{w}{2} a^2\).

Step 4: Calculate the Average Pressure

Now find the average pressure by dividing the result of the integration by \(a\): \[\frac{1}{a} \cdot \frac{w}{2} a^2 = \frac{w}{2} a\]. This simplifies to \(\frac{w a}{2}\).

Step 5: Conclusion and Interpretation

The average pressure along the vertical dimension of the plate is \(\frac{w a}{2}\). This result represents the mean pressure acting on the plate due to its complete submersion in the fluid.

Key Concepts

pressure and fluid mechanics

Pressure in fluid mechanics is a fundamental concept that describes how fluids (liquids and gases) exert force on objects submerged within them. This force is caused by the weight of the fluid above the object, compressed by gravity. The deeper you go in a fluid, the greater the pressure, because there is more fluid above and thus more weight. Pressure is calculated using the formula:

• \( P(y) = w \cdot y \)

Here, \(w\) is the weight-density of the fluid, and \(y\) is the depth in the fluid. This simple relation shows that pressure increases linearly with depth.

integration in calculus

Integration is a powerful tool in calculus that enables us to accumulate quantities, like finding areas under curves or, in this case, calculating total pressure over an area. When determining average pressure on the submerged plate, we integrate the expression for pressure over the depth of the plate:

• \( \int_0^a P(y) \, dy \)
• \( \int_0^a w \, y \, dy \)

The integral results in calculating the total pressure applied over the height of the plate. Solving this leads to a constant multiplied by the depth squared term, indicating the sum of pressures across all depths. This integral process allows us to account for all the slight increases in pressure as the depth increases.

weight-density concept

The concept of weight-density is key in fluid mechanics and refers to how heavy a fluid is per unit volume. It reflects the gravitational pull on the fluid's mass, affecting the pressure exerted at any depth. Weight-density \(w\) is crucial for calculating pressure, as it directly scales with how much pressure a fluid column imposes.

• Weight-density allows for easy modeling of how fluids behave.
• It is measured as force per unit volume, typically in units like pounds per cubic foot or Newtons per cubic meter.

A higher weight-density means more pressure will be felt at a given depth, which is essential to understand when modeling fluid-related problems.

mathematical modeling in physics

Mathematical modeling is the process of using mathematical structures and concepts to represent real-world systems. In this exercise, we're modeling the pressure exerted on a submerged plate through finely-tuned mathematical expressions and calculus.

• The formula \( P(y) = w \cdot y \) is a model showing how pressure varies with depth.
• Integration further refines this model by accounting for total effects over a vertical plate.

Modeling helps in predicting how systems behave under various conditions, like submersion depths or changes in weight-density, enabling engineers and scientists to design safe and efficient systems in fluid-related environments.