Question

A tuning fork of frequency $480\mathrm{Hz}$ produces 10 beats/s when sounded with a vibrating sonometer string. What must have been the frequency of the string if a slight increase in tension produces fewer beats per second than before? (A) $480\mathrm{Hz}$ (B) $490\mathrm{Hz}$ (C) $460\mathrm{Hz}$ (D) $470\mathrm{Hz}$

Short answer

The frequency of the sonometer string is 490 Hz (Option B).

Step-by-step solution

Write down the known variables and formula for beat frequency

We know the following variables: - Frequency of the tuning fork, $f1=480$ Hz - Beat frequency, $bf=10$ beats/s Now we need to find the frequency of the sonometer string, let's denote it by $f2$. The formula for beat frequency is: $bf=|f1-f2|$

Use the given information to determine the possible values for the string's frequency

We know that if the tension is increased, the beat frequency decreases, so we have two possibilities: 1. The tuning fork's frequency is higher than the string's frequency, and increasing the tension makes the string's frequency closer to the fork's frequency. 2. The tuning fork's frequency is lower than the string's frequency, and increasing the tension makes the string's frequency closer to the fork's frequency. Since the beat frequency must decrease when the tension increases, we can rule out the first possibility: if the tuning fork's frequency is higher than the string's frequency, increasing the tension would move string's frequency away from the fork's frequency. That leaves us with the second possibility: the tuning fork's frequency is lower than the string's frequency. Now we'll use the formula for beat frequency: $bf=|f1-f2|$ $10=|480-f2|$

Solve for the string's frequency

We'll solve the equation for $f2$: $f2=480+10$ or $f2=480-10$ Based on our analysis, we determined that the tuning fork's frequency is lower than the string's frequency, so we can use the first equation: $f2=480+10=490$ Hz Therefore, the frequency of the sonometer string is 490 Hz, which corresponds to option (B).

Key concepts

Beat Frequency

Beat frequency is a fascinating concept in sound waves that occurs when two musical notes, having slightly different frequencies, are played together. The result is a periodic variation in sound intensity, known as beats. You hear these variations as a "wobbling" sound.
Understanding beat frequency is crucial in music and physics. Here's a simple breakdown:

• The beat frequency is the absolute difference between the two frequencies you are comparing. You can calculate it using the formula: $bf=|f1-f2|$, where $bf$ is the beat frequency, $f1$ is the frequency of the first source, and $f2$ is the frequency of the second source.
• Audibly, a lower beat frequency means the sound seems smoother, while a higher beat frequency makes the sound vibrate more rapidly.

In practical applications, tuning instruments or diagnosing sound equipment often involves analyzing beats to achieve perfect harmony.

Tuning Fork

A tuning fork is a simple device yet incredibly effective for producing specific tones. It typically consists of a U-shaped metal bar that, when struck, vibrates at a particular frequency. This property makes it invaluable in musical instrument tuning and acoustic experiments.
Key attributes of a tuning fork:

• It produces a pure and constant sound frequency, ideal for setting a reference tone.
• Its simplicity ensures minimal distortion, providing clear and specific pitches.
• For frequencies beyond the reach of human-made devices, tuning forks serve as a foundational tool.

The fork's frequency is determined by its size and the material it's made from. For example, striking a 480 Hz tuning fork vibrates the air at a frequency of 480 times per second. It assists in determining unknown frequencies by matching its sound with another source and observing the resulting beat frequency.

Sonometer

A sonometer is an apparatus used to study the frequency of vibrating strings. It consists of a hollow wooden box that amplifies sound, with a stretched wire over it. Weights adjust the tension on the wire, influencing its frequency. How a sonometer works:

• Changing the wire's tension alters its vibration frequency. An increase in tension typically increases the frequency.
• By using sliding bridges, you can change the effective length of the string, affecting its frequency, similar to how a guitar string functions.
• It is often used with a tuning fork to understand and calculate unknown frequencies via the phenomenon of resonance and beat frequency.

The sonometer is a valuable educational tool for demonstrating the principles of sound waves and vibrations, making the abstract concept of frequency visible and audible.

Frequency Calculation

Calculating frequency is crucial when dealing with sound waves. It's the measure of how often a sound wave repeats in a unit of time, usually expressed in hertz (Hz).The formula for frequency calculation depends on the context:

• For sound, it usually involves determining how many times the wave cycles per second.
• With musical instruments or devices like tuning forks and sonometers, you often calculate unknown frequencies by analyzing their interaction with a known reference frequency.

In exercises like the one presented, understanding frequency calculation allows you to solve for the unknown frequency of a string when combined with a tuning fork. You use the formula $bf=|f1-f2|$ and given values to find such unknowns. Understanding this concept allows you to harmonize sounds and solve complex mathematical relationships between different pitches. It's essential for both musical tuning and the study of sound dynamics.