Question

A tuning fork of frequency \(200 \mathrm{~Hz}\) is in unison with a sonometer wire. The number of beats heard per second when the tension is increased by \(1 \%\) is \(\ldots \ldots \ldots .\) (A) 1 (B) 2 (C) 4 (D) \(0.5\)

Short answer

The number of beats heard per second when the tension is increased by 1% is 1 (option A).

Step-by-step solution

Find the initial frequency of the sonometer wire

Since the tuning fork and sonometer wire are initially in unison, their frequencies are the same. Therefore, the initial frequency of the sonometer wire, \(f_1\), is 200 Hz.

Determine the new tension of the sonometer wire

We know that the tension in the wire has been increased by 1%. Let the initial tension be \(T\). Therefore, the new tension, \(T'\), can be found using the following relation: \[T' = T + 0.01T\]

Calculate the change in frequency due to the change in tension

The frequency of a sonometer wire is related to its tension by the following equation: \(f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}\) where \(f\) is the frequency, \(L\) is the length of the wire, and \(\mu\) is the linear density of the wire. When the tension changes, the new frequency, \(f'\), can be found using the same equation: \(f' = \frac{1}{2L}\sqrt{\frac{T'}{\mu}}\)

Solve for the new frequency, \(f'\)

We need to find the ratio of the new frequency to the old frequency to find the new frequency. Using the equation for \(f\) and \(f'\) above and dividing the equations, we get: \(\frac{f'}{f} = \frac{\sqrt{T'}}{\sqrt{T}}\) Since \(f = 200 \, Hz\), we can find the new frequency \(f'\) using the new tension, \(T'\): \(f' = 200 \, Hz \cdot \frac{\sqrt{T'}}{\sqrt{T}} = 200 \, Hz \cdot \sqrt{\frac{T'}{T}}\) Substitute the equation for the new tension, \(T'\), from Step 2: \(f' = 200 \, Hz \cdot \sqrt{\frac{T + 0.01T}{T}} = 200 \, Hz \cdot \sqrt{1 + 0.01}\)

Calculate the number of beats heard per second

The beat frequency can be found by taking the difference between the new frequency, \(f'\), and the initial frequency, \(f\): Beat frequency = \(f' - f\) Using the equation for \(f'\) found in Step 4: Beat frequency = \(200 \, Hz \cdot \sqrt{1 + 0.01} - 200 \, Hz\) After evaluating the expression, we find that the beat frequency is approximately 1 Hz. Therefore, the number of beats heard per second when the tension is increased by 1% is 1 (option A).

Key concepts

Tuning Fork

A tuning fork is a simple device used to produce a specific and consistent tone. Its main component is a U-shaped metal bar that vibrates when struck, creating sound waves. This vibration generates a precise pitch, known as frequency, which is determined by the fork's material and dimensions.

• The pitch or frequency of a tuning fork does not change unless the fork itself is physically altered.
• This consistent frequency makes it an excellent tool for calibrating musical instruments and in scientific experiments.

When used with a sonometer, a tuning fork helps establish what frequency the sonometer wire should be at for them to be in unison. This is similar to using a reference point to ensure accuracy in sound production.

Frequency

Frequency refers to how many times a cycle or wave occurs in one second, measured in Hertz (Hz). In a musical sense, it represents how many vibrations a sound wave undergoes each second.
For example, a tuning fork labeled as 200 Hz implies the fork vibrates 200 times per second.

• Frequency directly correlates with the pitch of a sound; higher frequencies correspond to higher pitches.
• Frequency can be affected by the medium through which the sound travels or any physical changes to the sound source.

In a sonometer wire, the frequency is influenced by factors like tension, length, and linear density. When a change in tension occurs, the sound wave frequency produced by the wire also changes.

Beat Frequency

Beat frequency occurs when two sound waves of slightly different frequencies interfere with one another. This interference results in a new frequency, heard as pulsations or "beats."
To determine beat frequency:

• Subtract the frequency of one wave from the other.
• The absolute difference gives the number of beats per second.

For instance, if you have a tuning fork of 200 Hz and a sonometer wire that slightly shifts in frequency due to tension change, beats occur. The problem demonstrates this concept with a tuning fork hitting 200 Hz and the sonometer wire's frequency changing slightly, resulting in beats heard by a listener.

Tension and Frequency Relationship

The tension in a string or wire, like in a sonometer, directly affects frequency. According to the fundamental physics formula, frequency is proportional to the square root of the tension divided by the linear density of the string.

• This means increasing the tension will increase the frequency.
• Conversely, decreasing tension leads to a lower frequency.

In practical terms, adjusting the tension often involves tightening or loosening a wire or string. Even a small change, like a 1% increase in tension, alters the frequency slightly, as demonstrated in the original problem. The formula explains why increasing the tension by 1% results in a new frequency that, when compared with the original, produces a noticeable beat frequency. This is crucial for musicians and physicists alike, as tiny adjustments can bring about significant audible differences.