Problem 47 - Precalculus Mathematics for Calculus

Question

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

Step 1: Understand the Relationship

The problem states that the frequency \( f \) is inversely proportional to the length \( L \) of the violin string. Inverse proportionality means that as one quantity increases, the other decreases at a constant rate.

Step 2: Write the Equation for Inverse Proportionality

An inverse proportionality can be written using a constant of proportionality \( k \). Thus, the relationship can be modeled by:\[f = \frac{k}{L}\]This equation shows that the frequency \( f \) is the constant \( k \) divided by the length \( L \).

Step 3: Analyze the Effect of Doubling the Length

If the length \( L \) is doubled, consider the new length as \( 2L \). Substitute this into the equation for frequency:\[f' = \frac{k}{2L}\]Compare this new frequency \( f' \) to the original frequency \( f \):\[f' = \frac{1}{2} \times \frac{k}{L} = \frac{1}{2}f\]This shows that doubling the length results in halving the frequency.

Key Concepts

Frequency of Vibration

The frequency of vibration of a violin string refers to how many times the string oscillates back and forth in one second. It's denoted by the symbol \( f \). Frequency is a critical factor in determining the pitch of the sound produced by the string.

Understanding frequency involves recognizing it as a measure of vibration cycles. A higher frequency means a higher pitch, whereas a lower frequency corresponds to a lower pitch. In musical terms, different frequencies correspond to different notes you hear from an instrument.

• Higher frequency = higher pitch
• Lower frequency = lower pitch

In the context of a violin string, this concept ties closely to the string's physical properties, like its tension, length, and density. As you'll see, improving or changing these elements alters the frequency and thus the sound produced.

Violin String

A violin string is a critical component of the instrument, made to produce sound through vibrations. It is typically constructed from materials like gut, steel, or synthetic fibers, each having unique properties influencing sound production.

Several factors affect how a violin string vibrates and thus the frequency of sound it produces:

• Material: Different string materials can produce different tones.
• Tension: The string must be tightly enough to vibrate effectively, altering tension can change the frequency.
• Density: Defines mass per unit volume of the string, which impacts frequency.

When a player draws a bow across the string or plucks it, the string vibrates, producing sound. Adjustments in the string's tension and tuning change the frequency, allowing the musician to play different notes.

Length of a String

The length of a string on a violin plays a crucial role in determining its frequency of vibration. Longer strings generally produce lower frequencies, while shorter strings create higher frequencies. This inverse relationship is because as the length increases, there is more material to vibrate, which slows the vibration rate.

In mathematical terms, this is expressed as \( f = \frac{k}{L} \). Here, \( f \) is the frequency, \( L \) is the length of the string, and \( k \) is a constant that depends on the tension and density of the string.

• If you increase the length \( L \), the frequency \( f \) decreases.
• Doubling the length of the string, for example, halves the frequency, which makes the sound produced lower in pitch.

Understanding this relationship helps musicians and designers of musical instruments to manipulate the sound and pitch produced by string instruments effectively. Mastery of length adjustments means better control over the music played.