Problem 42 - Algebra Form and Function

Question

The thrust, \(T,\) in pounds, of a ship's propeller is proportional to the square of the propeller speed, \(R\), in rotations per minute, times the fourth power of the propeller diameter, \(D,\) in feet. \(^{8}\) (a) Write a formula for \(T\) in terms of \(R\) and \(D\). (b) Solve for the propeller speed \(R\) in terms of the thrust \(T\) and the diameter \(D\). Write your answer in the form \(R=C T^{n} D^{m}\) for some constants \(C\), \(n,\) and \(m\). What are the values of \(n\) and \(m\) ? (c) Solve for the propeller diameter \(D\) in terms of the thrust \(T\) and the speed \(R\). Write your answer in the form \(D=C T^{n} R^{m}\) for some constants \(C, n,\) and \(m\). What are the values of \(n\) and \(m\) ?

Step-by-Step Solution

Answer

Question: The thrust (T) of a ship's propeller is proportional to the square of the propeller speed (R) in rotations per minute, times the fourth power of the propeller diameter (D) in feet. Find the values n and m in the following expressions: \(R = C*T^{n}*D^{m}\) and \(D = C'*T^{n'}*R^{m'}\). Answer: For the expression \(R = C*T^{n}*D^{m}\), the values of n and m are \(\frac{1}{2}\) and \(-2\), respectively. For the expression \(D = C'*T^{n'}*R^{m'}\), the values of n' and m' are \(\frac{1}{4}\) and \(-\frac{1}{2}\), respectively.

Step 1: Write the formula for thrust, T

Since the thrust T is proportional to the square of the propeller speed R and the fourth power of the propeller diameter D, we can write the formula as: \(T=kR^{2}D^{4},\) where k is the constant of proportionality.

Step 2: Solve for R in terms of T and D

We want to find the formula for \(R\) in terms of \(T\) and \(D\). To do this, we can isolate \(R\) in the formula we just wrote by dividing both sides by \(kD^{4}\), we get: \(R^{2} = \frac{T}{kD^{4}}\) Now take the square root of both sides: \(R =\sqrt{\frac{T}{kD^{4}}}\) We can rewrite this as: \(R = C*T^{n}*D^{m}\) Where \(C = \frac{1}{\sqrt{k}}\), \(n = \frac{1}{2}\) and \(m = -2\). So the values of \(n\) and \(m\) are \(\frac{1}{2}\) and \(-2\) respectively.

Step 3: Solve for D in terms of T and R

Now we want to find the formula for propeller diameter \(D\) in terms of thrust \(T\) and speed \(R\). To do this, we can use the same equation for thrust, divide both sides by \(kR^{2}\) and take the fourth root of both sides: \(D^{4} = \frac{T}{kR^{2}}\) Now take the fourth root of both sides: \(D = \sqrt[4]{\frac{T}{kR^{2}}}\) We can rewrite this as: \(D = C*T^{n}*R^{m}\) Where \(C = \frac{1}{\sqrt[4]{k}}\), \(n = \frac{1}{4}\) and \(m = -\frac{1}{2}\). So the values of \(n\) and \(m\) are \(\frac{1}{4}\) and \(-\frac{1}{2}\), respectively.

Key Concepts

Proportionality

Proportionality is a simple yet crucial concept in physics and math. It describes how one quantity changes with respect to another. If two quantities are proportional, this means when one changes, the other changes in a consistent way, either increasing or decreasing in a predictable manner.

In the context of thrust calculation for propellers, the thrust, denoted as \(T\), is directly proportional to specific powers of the propeller's speed \(R\) and its diameter \(D\). This means:

• An increase in propeller speed \(R\) leads to an increase in thrust \(T\), assuming the diameter \(D\) remains constant.
• Similarly, increasing the diameter \(D\) while keeping the speed \(R\) constant also results in a higher thrust \(T\).

This relationship allows for formula creation that reflects how such variables are interconnected. The equation for thrust determined by proportionality is \(T = kR^{2}D^{4}\), where \(k\) is the constant of proportionality. This means that thrust increases with the square of the speed and the fourth power of the diameter.

Propeller Speed

Propeller speed \(R\) is an essential component in calculating thrust. This is because the speed at which a propeller rotates impacts how much force it can exert on the water, thereby affecting the vessel's ability to move.

When solving for \(R\) in terms of \(T\) (thrust) and \(D\) (diameter), understanding the relationship through proportionality becomes critical. From our initial equation \(T=kR^2D^4\), we rearrange to solve for \(R\) and find that:

• \(R = \sqrt{\frac{T}{kD^4}}\).

This formula indicates that an increase in thrust usually requires an increase in speed, especially if diameter remains constant. Additionally, from the formula transformation into \(R = C*T^{n}*D^{m}\), where \(C, n,\) and \(m\) are defined constants, we learn that:

• \(n = \frac{1}{2}\)
• \(m = -2\)

These exponents further allow us to understand how speed reacts to changes in thrust and diameter.

Propeller Diameter

The propeller diameter \(D\) significantly influences the thrust a propeller can produce. A larger diameter generally means a propeller can move more water, resulting in higher force and speed capabilities.

When exploring how \(D\) relates to thrust \(T\) and propeller speed \(R\), you can see this by rearranging our primary formula to solve for \(D\). From \(T=kR^2D^4\), we derive:

• \(D = \sqrt[4]{\frac{T}{kR^2}}\).

This equation shows that diameter must increase as thrust increases, given a constant speed. Converting this into the format of \(D = C*T^{n}*R^{m}\), where constants \(C, n, m\) provide more insights, we identify:

• \(n = \frac{1}{4}\)
• \(m = -\frac{1}{2}\)

These exponents portray how diameter balances between varying thrust and speed, offering more design flexibility.

Mathematical Formulation

Mathematical formulation is the process of translating a physical situation into a mathematical equation. This translation allows us to use math to predict, analyze, and understand the interaction. In our exercise, the problem defined the relationship between thrust, propeller speed, and diameter.

Starting from the concept of proportionality, we derived the foundational equation for thrust \(T = kR^2D^4\). This formula indicates not only the relationships but how to solve for one variable when the others are known. Through mathematical formulation:

• We determined how to equate \(R\) and \(D\) in terms of \(T\).
• The relationships are accommodated as \(R = \sqrt{\frac{T}{kD^4}}\) and \(D = \sqrt[4]{\frac{T}{kR^2}}\).

Converting these into general proportional forms like \(R = C*T^{n}*D^{m}\) and \(D = C*T^{n}*R^{m}\) assists in visualizing how thrust, speed, and diameter are interlinked by constants \(C, n, m\). This elegant mathematical encapsulation allows engineers and designers to tailor and tweak propeller designs effectively, depending on specific requirements.