Question

The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 48 inches long and vibrates 256 times per second, what is the length of a string that vibrates 576 times per second?

Short answer

The new length of the string is 21.33 inches.

Step-by-step solution

- Understand the relationship

The rate of vibration varies inversely with the length of the string. This means that as the length of the string increases, the rate of vibration decreases, and vice versa. Mathematically, this can be represented as: \[ f \times l = k \] where \( f \) is the frequency, \( l \) is the length of the string, and \( k \) is a constant.

- Determine the constant

Using the given values for the first string, substitute \( f = 256 \) vibrations per second and \( l = 48 \) inches into the equation \[ 256 \times 48 = k \]Calculate the constant \( k \): \[ k = 256 \times 48 = 12288 \]

- Find the new length

With the constant \( k \) determined, use the given frequency of the new string, \( f = 576 \) vibrations per second. Substitute \( f = 576 \) and \( k = 12288 \) into the equation to find the new length \( l \): \[ 576 \times l = 12288 \] Solve for \( l \): \[ l = \frac{12288}{576} = 21.33 \text{ inches} \]

Key concepts

Rate of Vibration

The rate of vibration refers to how frequently something oscillates or moves back and forth. In the context of a vibrating string, this is typically measured in vibrations per second, also known as frequency.
When considering the rate of vibration, it is essential to understand that different factors can influence it. In our example with the vibrating string, the length of the string plays a significant role.
As a general rule:

• If the string is longer, it will vibrate more slowly, thus having a lower frequency.
• If the string is shorter, it will vibrate more quickly, thus having a higher frequency.

Understanding this fundamental concept helps you grasp why changing the length of the string affects how fast it vibrates.

Constant of Variation

In inverse variation problems, you'll frequently encounter something called the 'constant of variation,' denoted as k.
In this specific exercise, the equation is \( f \times l = k \), where f is the frequency and l is the length. This signifies that when you multiply the frequency by the length of the string, you get a constant value.
Let's break down the steps to identify the constant of variation:

• You start with the given values. Here, f = 256 vibrations per second, and l = 48 inches.
• Substitute these into the equation to find k: 256 \times 48 = 12288
• So, our constant of variation k equals 12288. This value remains constant as long as the tension in the string remains unchanged.

Understanding the constant of variation is crucial, as it allows you to find unknown variables when given different sets of frequency and length.

Frequency and Length Relationship

One of the core ideas in this exercise is understanding the relationship between frequency and length. This concept stems from the idea of inverse variation.
Inverse variation implies that when one quantity increases, the other decreases proportionally. In mathematical terms, we use the formula: \( f \times l = k \)
With this relationship in mind:

• If you increase the length (l), the frequency (f) will decrease to maintain the constant k.
• If you decrease the length (l), the frequency (f) will increase.

This relationship helps you to immediately see the balance. In our exercise, after finding the constant k, we could determine the new length of the string by rearranging the equation:

• Given new frequency f = 576 vibrations per second
• We used 12288 = 576 \times l
• Solving for l, we found the new length to be 21.33 inches.

This relationship is vital for solving similar problems regarding the inverse variation of quantities.

Algebraic Equations

Algebraic equations are essential tools in solving problems involving variable relationships. In this exercise, understanding how to manipulate the algebraic equation was key to finding the solution.
We started with the basic equation of inverse variation: \( f \times l = k \)
Here's a step-by-step approach for using algebraic equations:

• Identify the given values and substitute them into the equation to find the constant, k. For example, 256 \times 48 = 12288.
• Use the constant in conjunction with the new given value to find the unknown variable. For example, using k = 12288 and f = 576 vibrations per second, we can solve for the new length l.
• Solve the equation. Rearrange the equation to isolate the unknown variable and perform the calculations: \[ l = \frac{12288}{576} = 21.33\text{ inches} \]

Algebraic equations serve as the backbone for solving such problems. Understanding the steps and keeping the variables in mind will help you efficiently find the solutions.