Problem 47 - Algebra and Trigonometry

Question

Frequency of Vibration The frequency \(f\) of vibration of a violin string is inversely proportional to its length \(L\) . The constant of proportionality \(k\) is positive and depends on the tension and density of the string. (a) Write an equation that represents this variation. (b) What effect does doubling the length of the string have on the frequency of its vibration?

Step-by-Step Solution

Answer

(a) \(f = \frac{k}{L}\); (b) Doubling the length halves the frequency.

Step 1: Understand Proportional Relationships

When a quantity, such as frequency \(f\), is inversely proportional to another quantity, such as length \(L\), it means that as one increases, the other decreases. The relationship can be expressed with a proportionality constant \(k\).

Step 2: Write the Inverse Proportionality Equation

Since frequency \(f\) is inversely proportional to the length \(L\), we can write \(f = \frac{k}{L}\), where \(k\) is a positive constant that depends on the tension and density of the string.

Step 3: Analyze the Effect of Doubling the Length

If the length \(L\) of the string is doubled, the new length is \(2L\). Substitute \(2L\) into the equation: \(f_{new} = \frac{k}{2L}\).

Step 4: Compare Frequency Before and After Doubling

Originally, \(f = \frac{k}{L}\). With the doubled length, \(f_{new} = \frac{k}{2L}\). This simplifies to \(f_{new} = \frac{1}{2} \times \frac{k}{L} = \frac{f}{2}\). The new frequency is half the original frequency.

Key Concepts

Frequency of Vibration

The frequency of vibration of a violin string refers to how often the string vibrates per second and is measured in Hertz (Hz).
This concept is pivotal to understanding how musical sounds are produced, as the frequency determines the pitch of the note the string produces. Imagine the waves generated by the vibrating string - the more waves per second, the higher the pitch or the frequency.
Think of frequency like the heartbeat of your music; it pulses at a steady rate, creating the rhythm and melody.

• Higher frequencies produce higher pitches.
• Lower frequencies result in deeper, richer tones.

Given its role, understanding how other factors affect frequency is vital. For a violin, other factors like the string's length significantly influence the frequency.

Violin String Length

The length of a violin string significantly impacts its frequency of vibration. When we talk about string length, we're discussing the distance between the two points where the string is fixed.
In the context of inverse proportionality, as this length increases, the frequency of vibration decreases.
This means longer strings vibrate more slowly, which results in a lower pitch.

• Doubling the length of the string results in halving the frequency, producing a pitch an octave lower.
• Shortening the string results in a higher frequency, raising the pitch.

Understanding this relationship helps musicians and instrument makers control the pitch and tone of different instruments by adjusting string length.

Proportionality Constant

In this context, a proportionality constant is a steady value that makes the equation of inverse proportionality work.
This constant, denoted as \(k\), depends on factors like the tension and density of the string.
In the equation \(f = \frac{k}{L}\), \(k\) ensures that the frequency and length are correctly related.

• Higher tension increases \(k\), leading to higher frequencies.
• Greater density reduces \(k\), which results in lower frequencies.

The proportionality constant ties physical properties of the string to its vibrational characteristics. Understanding \(k\) enables better predictions about the sound a string will produce under varying conditions.