Problem 30 - Applied Calculus

Question

The power \(P\) produced by a windmill is proportional to the square of the diameter \(d\) of the windmill and to the cube of the speed \(v\) of the wind. \(^{5}\) (a) Write a formula for \(P\) as a function of \(d\) and \(v\) (b) A windmill generates \(100 \mathrm{kW}\) of power at a certain wind speed. If a second windmill is built having twice the diameter of the original, what fraction of the original wind speed is needed by the second windmill to produce \(100 \mathrm{kW} ?\) (c) Sketch a contour diagram for \(P\).

Step-by-Step Solution

Answer

(a) \(P = k \cdot d^2 \cdot v^3\). (b) Speed fraction is \(\frac{1}{\sqrt[3]{4}}\) of original. (c) Contours show \(d^2\) vs. \(v^3\) for constant \(P\).

Step 1: Define the Proportionality Formula

Since the power \(P\) is proportional to the square of the diameter \(d\) and the cube of the speed \(v\), we can express this relationship as \(P = k \cdot d^2 \cdot v^3\), where \(k\) is a proportionality constant.

Step 2: Express Diameter and Speed Relationship

From the formula \(P = 100 \text{kW} = k \cdot d^2 \cdot v^3\), let's consider a second windmill with twice the original diameter, making it \(d' = 2d\). We need to determine the fractional speed \(v'\) that satisfies the equation: \(100 = k \cdot (2d)^2 \cdot v'^3\). Thus, \(100 = 4 \cdot k \cdot d^2 \cdot v'^3\).

Step 3: Solve for New Speed

We have the equations \(k \cdot d^2 \cdot v^3 = 100\) and \(4 \cdot k \cdot d^2 \cdot v'^3 = 100\). Simplify to find \(v'^3 = \frac{v^3}{4}\). Taking the cube root, we get \(v' = \frac{v}{\sqrt[3]{4}} \approx \frac{v}{1.5874}\). This means the new speed \(v'\) is \(\frac{1}{\sqrt[3]{4}}\) of the original speed.

Step 4: Sketch a Contour Diagram

To sketch a contour diagram for \(P\), plot curves on a graph where each curve represents a constant value of \(P\). On the axes, represent \(d\) and \(v\). The formula \(P = k \cdot d^2 \cdot v^3\) indicates that higher contours (constant \(P\) values) move away from the origin, illustrating that increased diameter or wind speed increases power generation.

Key Concepts

Proportionality

Proportionality is a fundamental concept in mathematics and physics, describing a relationship between two variables where one is a constant multiple of the other. In the context of this problem, the power output \( P \) of a windmill is directly proportional to both the square of its diameter \( d \) and the cube of the wind speed \( v \). This means:

• When the diameter doubles, the power increases by four times.
• When the wind speed doubles, the power increases by eight times, since \(2^3 = 8\).

This is expressed mathematically as:\[ P = k \cdot d^2 \cdot v^3 \]\( k \) is a constant of proportionality that remains unchanged regardless of \( d \) and \( v \). The role of \( k \) is crucial, as it adjusts the units of measurement to fit real-world data.

Power and Energy Calculations

Calculating power output for a device such as a windmill involves understanding the variables at play. Here, the windmill's power \( P \) depends on both the diameter \( d \) and wind speed \( v \). For example, given that a windmill generates \(100 \text{kW}\) at a specific speed, we can explore changes in diameter and how they affect the required wind speed to maintain the same power output.

• Initially, the formula \( P = k \cdot d^2 \cdot v^3 = 100 \) represents this known state.
• Doubling the diameter leads to a new formula: \( 100 = 4 \cdot k \cdot d^2 \cdot v'^3 \).
• Solving for \( v' \), we derive \( v' = \frac{v}{\sqrt[3]{4}} \) to maintain power output.

This calculation shows that by adjusting the diameter of the windmill, less wind speed is needed to produce the same power, demonstrating an inherent efficiency gained from a larger size.

Contour Diagrams

Contour diagrams are visual tools that help illustrate relationships between variables by representing lines of constant value. In this context, they're used to show how power \( P \) remains constant for varying diameters \( d \) and wind speeds \( v \).

• Each contour line on the graph represents a constant power value.
• The axes would typically display \( d \) and \( v \), while each line shows combinations that yield the same \( P \).
• As diameter or speed increases, the contours move further from the origin, showcasing the increased power generation capability.

Contour diagrams simplify complex multi-variable relationships by allowing for quick visualization of how changing one parameter affects the system. They provide a clear way to understand the balance needed between diameter and wind speed to maintain or achieve specific power outputs.