Problem 57 - Basic Technical Mathematics with Calculus

Question

Set up the appropriate quadratic equations and solve. The spring constant \(k\) is the force \(F\) divided by the amount \(x\) the spring stretches \((k=F / x) .\) See Fig. \(7.2(a)\) For two springs in series [see Fig. \(7.2(\mathrm{b})],\) the reciprocal of the spring constant \(k_{c}\) for the combination equals the sum of the reciprocals of the individual spring constants. Find the spring constants for each of two springs in series if \(k_{c}=2 \mathrm{N} / \mathrm{cm}\) and one spring constant is \(3 \mathrm{N} / \mathrm{cm}\) more than the other.

Step-by-Step Solution

Answer

The spring constants are 3 N/cm and 6 N/cm.

Step 1: Set up the Equation for Springs in Series

For two springs in series, the formula for the combined spring constant is given by \( \frac{1}{k_c} = \frac{1}{k_1} + \frac{1}{k_2} \), where \( k_c \) is the combined spring constant, and \( k_1 \) and \( k_2 \) are the spring constants of the individual springs.

Step 2: Equation with Given Constants

We know that \( k_c = 2 \, \text{N/cm} \) and \( k_2 = k_1 + 3 \). Substitute these into the equation: \( \frac{1}{2} = \frac{1}{k_1} + \frac{1}{k_1 + 3} \).

Step 3: Clear the Fractions

To clear the fractions, multiply both sides by \( 2k_1(k_1 + 3) \): \[ k_1(k_1 + 3) = 2(k_1 + 3) + 2k_1 \].

Step 4: Simplify the Equation

Expand the equation: \[ k_1^2 + 3k_1 = 2k_1 + 6 + 2k_1 \].Combine like terms to get:\[ k_1^2 - k_1 - 6 = 0 \].

Step 5: Solve the Quadratic Equation

We solve the equation \( k_1^2 - k_1 - 6 = 0 \) using the quadratic formula: \[ k_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \],with \( a = 1 \), \( b = -1 \), and \( c = -6 \).

Step 6: Calculate the Roots

Calculate the discriminant: \[ b^2 - 4ac = (-1)^2 - 4(1)(-6) = 1 + 24 = 25 \].Then, calculate each root:\[ k_1 = \frac{1 \pm 5}{2} \].So, \( k_1 = 3 \) or \( k_1 = -2 \). Since a spring constant must be positive, we choose \( k_1 = 3 \).

Step 7: Determine the Second Spring Constant

Given that \( k_2 = k_1 + 3 \), substitute \( k_1 = 3 \):\[ k_2 = 3 + 3 = 6 \].

Key Concepts

Understanding the Spring Constant

The spring constant, often represented by the symbol \( k \), is a fundamental concept related to the stiffness of a spring. It relates to the force applied to stretch or compress the spring and the distance it stretches or compresses. This is mathematically expressed as \( k = \frac{F}{x} \), where \( F \) is the force applied, and \( x \) is the displacement from the spring's original position.
It's important to note that the spring constant is unique to each spring and depends on factors such as the material of the spring and its dimensions.

• For a given force, a spring with a larger spring constant will stretch less than one with a smaller spring constant.
• The unit of spring constant is Newton per centimeter (N/cm), indicating how much force is needed to stretch the spring by one centimeter.

This concept is applied widely not only in theoretical physics problems but also in real-world engineering scenarios like car suspensions and measuring instruments.

Springs in Series

When two or more springs are arranged in series, they effectively behave differently than when isolated. For such a setup, the combined spring constant \( k_c \) is calculated using the reciprocals of the individual spring constants \( k_1 \) and \( k_2 \). The formula is given by \( \frac{1}{k_c} = \frac{1}{k_1} + \frac{1}{k_2} \).
This relationship arises because the total displacement of springs in series is the sum of individual displacements, and they each experience the same force.

• This setup may be used when the individual springs need to bear more combined load over a longer displacement.
• In practical applications, series springs can be found in various machinery and equipment requiring flexibility and adaptability.

By understanding this concept, we can better analyze and solve systems involving multiple springs, such as the example problem where springs have differing stiffness due to varying spring constants.

Solving with the Quadratic Formula

The quadratic formula is a powerful tool used to solve equations of the form \( ax^2 + bx + c = 0 \). In the context of resolving spring constants in series, the quadratic formula was employed in determining an unknown spring constant.
The formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here's how it works:

• Calculate the discriminant \( b^2 - 4ac \). This helps in determining the nature of the roots (real or complex).
• Calculate the roots using the quadratic formula by substituting the discriminant and the coefficients \( a \), \( b \), and \( c \).

In our example, the equation \( k_1^2 - k_1 - 6 = 0 \) was solved using the quadratic formula, resulting in two possible solutions: 3 or -2. Since spring constants cannot be negative, the correct solution was determined to be 3. Mastery of the quadratic formula is essential for tackling a variety of physics and mathematics problems.

Physics Problem Solving

Solving physics problems often involves setting up equations that describe physical systems accurately. As illustrated by the spring scenario, here's the approach:

• Understand the physical problem and translate it into mathematical language by identifying relevant equations and variables.
• Set up your equations based on the initial conditions and known relationships, like how the spring constant relationships were derived for springs in series.
• Apply mathematical techniques, such as algebraic manipulation or the quadratic formula, to solve the equations.
• Interpret the solutions critically. Ensure they make sense in a physical context, as done when rejecting the negative spring constant.

By following these clear, logical steps, students can effectively solve various physics problems. This practice cultivates critical thinking and enhances one's ability to tackle complex scenarios by breaking them into manageable parts.