Problem 71 - Intermediate Algebra

Question

Engineering. The stiffness \(k\) of the shaft shown is given by the formula $$ k=\frac{1}{\frac{1}{k_{1}}+\frac{1}{k_{2}}} $$ where \(k_{1}\) and \(k_{2}\) are the individual stiffnesses of each section. Simplify the complex fraction.

Step-by-Step Solution

Answer

The simplified stiffness is \( k = \frac{k_{1}k_{2}}{k_{1} + k_{2}} \).

Step 1: Understand the Formula

The given formula is for the equivalent stiffness of two sections in series. It is expressed as: \[ k = \frac{1}{\frac{1}{k_{1}} + \frac{1}{k_{2}}} \]. Our task is to simplify this complex fraction.

Step 2: Combine Denominators in the Sum

To simplify the denominator (\(\frac{1}{k_{1}} + \frac{1}{k_{2}}\)), find a common denominator, which is the product \(k_{1}\times k_{2}\). The expression becomes: \[ \frac{k_{2} + k_{1}}{k_{1}k_{2}} \].

Step 3: Simplify the Complex Fraction

Now the formula can be rewritten by substituting the simplified denominator into the overall expression for stiffness: \[ k = \frac{1}{\frac{k_{2} + k_{1}}{k_{1}k_{2}}} \]. This is equivalent to multiplying 1 by the reciprocal of the denominator: \[ k = \frac{k_{1}k_{2}}{k_{1} + k_{2}} \].

Key Concepts

Stiffness

Stiffness is a fundamental property in engineering that represents a material or component's ability to resist deformation under an applied force. It's commonly represented by the symbol \(k\) and measured in units of force per unit length (e.g., N/m). Understanding stiffness is crucial for engineers when designing structures and mechanical systems.

Key points about stiffness include:

• Stiffness quantifies how much a component or structure will deform under a given load. High stiffness means less deformation, while low stiffness indicates more deformation.
• In engineering applications, ensuring that all parts of a system have appropriate stiffness is critical for functionality and safety. For instance, in the design of a shaft, varying stiffness in different sections must be taken into account to ensure the overall component performs correctly under load.
• Stiffness is important in dynamic systems, where components must respond to loads that change over time, like in bridges or automotive suspension systems.

Complex Fractions

Complex fractions are fractions where either the numerator, denominator, or both contain a fraction themselves. Simplifying complex fractions is a common algebraic task in engineering mathematics, particularly when dealing with formulas that calculate properties like equivalent stiffness.

To simplify complex fractions:

• Identify all the fractional parts in the numerator and denominator.
• Combine fractions using a common denominator, if necessary, in either the numerator or denominator.
• Simplify by finding a reciprocal when necessary, which often turns a division into a multiplication with the reciprocal of the divisor.
• Ensure the end result is a single fraction in its simplest form, which aids in easier interpretation and further mathematical operations.

By simplifying complex fractions, engineers can derive clear and useful equations from intricate expressions. This step is essential when dealing with calculations of physical properties, ensuring that interpretations and further calculations are straightforward.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This process is vital for making calculations more manageable and reducing the risk of errors in engineering mathematics.

The key steps in algebraic simplification include:

• Identifying common factors in an expression that can be canceled out or combined to ease the overall equation.
• Using mathematical properties like distributive, associative, and commutative properties to rearrange and simplify terms.
• Performing operations such as combining like terms, reducing complex fractions, and factoring expressions to get a cleaner result.
• Ensuring dimensional accuracy, meaning the units of the simplified equation should reflect the intended physical property correctly.

Simplifying algebraic expressions is essential not only in theoretical calculations but also in practical engineering applications. This ensures equations are presented in the most efficient form possible, facilitating easier communication and application of technical concepts.