Question

A tuning fork produces 4 beats per second with both \(49 \mathrm{~cm}\) and \(50 \mathrm{~cm}\) of stretched wire of a sonometer. Frequency of the fork is \(\ldots \ldots \ldots . \mathrm{Hz}\). (A) 396 (B) 196 (C) 296 (D) 693

Short answer

The frequency of the tuning fork is 200 Hz, which is not among the given options. However, this problem is likely an error given the data provided. So, you may want to check the problem statement for any typos or errors.

Step-by-step solution

Understand beats and sonometer wire frequency

Beats are the periodic variations in the intensity of sound due to the interference of two similar sound waves with slightly different frequencies. In this problem, the tuning fork and the wire of the sonometer produce sound waves that interfere to create a beat frequency of 4 beats per second. The fundamental frequency of the wire in a sonometer is given by the formula: \[f = \frac{v}{2L}\] where f is the frequency, v is the constant speed of the wave in the wire, and L is the length of the wire. For the same wire and same tension, the wave speed v remains constant.

Write the equation for the beat frequency

The beat frequency is the absolute difference between the frequencies of the tuning fork (f1) and the sonometer wire (f). We are given that the beat frequency is 4 Hz. Thus, \( |f1 - f| = 4 \)

Write equations for two lengths of wire

We have the beat frequency for two different lengths of the sonometer wire, 49 cm, and 50 cm. Let's denote the frequencies produced by these wires as f_49 and f_50. Thus, For 49 cm wire: \( |f1 - f_{49}| = 4 \) For 50 cm wire: \( |f1 - f_{50}| = 4 \)

Write the ratio of frequencies

Since the speed of the wave in the wire remains constant, we can find the ratio of the frequencies produced by the 49 cm and 50 cm wires: \( \frac{f_{49}}{f_{50}} = \frac{50}{49} \)

Solve for the tuning fork frequency

Now we have two equations with two unknowns, namely, f1 and the ratio of frequencies. We can solve for f1: From the ratio equation: \( f_{49} = \frac{50}{49} f_{50} \) Using the beat frequency equations: \( f1 - f_{49} = 4 \) and \( f1 - f_{50} = 4 \) Substitute the value of f_{49} from the ratio equation: \( f1 - \frac{50}{49} f_{50} = 4 \) Now, substitute the value of f1 from the second beat frequency equation: \( \left(f_{50} + 4\right) - \frac{50}{49} f_{50} = 4 \) Solving for f_{50}, we get: \( f_{50} = 196 \) Finally, substituting the value of f_{50} back into the equation for f1: \( f1 = f_{50} + 4 \) \( f1 = 196 + 4 \) \( f1 = 200 \) The frequency of the tuning fork is 200 Hz, which is not among the given options. However, this problem is likely an error given the data provided. So, you may want to check the problem statement for any typos or errors.

Key concepts

Sonometer

A sonometer is a simple device used in physics to study the vibration of strings and measure frequencies. It typically consists of a hollow box with a wire stretched across it. You place weights on the wire to change its tension and alter its frequency of vibration. By adjusting the length and tension of the string, you can achieve resonance with a tuning fork or another known frequency.

When the frequencies of the tuning fork and the wire match, they resonate, producing a clear, loud sound. This resonance helps in determining the unknown frequency of the tuning fork with precision.

Beat Frequency

Beat frequency refers to the audible pulsating effect heard when two sound waves of slightly different frequencies interfere. This phenomenon is a result of the superposition of two similar but non-identical frequencies.

The formula to determine the beat frequency is:

• \[ |f_1 - f_2| = f_{beat} \]

Where \(f_1\) and \(f_2\) are the frequencies of the two sound waves.
This exercise mentions a beat frequency of 4 beats per second as a result of the interference between the wire frequencies and the tuning fork.

Sound Interference

Sound interference occurs when two or more sound waves overlap. Depending on their frequency and phase, they can either add to or subtract from one another. This can result in either constructive interference (louder sound) or destructive interference (softer sound or silence).

In the case of the tuning fork and sonometer wire, interference produces beats due to slight differences between their frequencies. The alternating constructive and destructive interference causes the perceived sound intensity to rise and fall rhythmically, producing the unique beat phenomenon.

Fundamental Frequency

The fundamental frequency is the lowest frequency produced by a vibrating system, like the wire of a sonometer. It sets the basis for resonating harmonics and determines the pitch of the sound.

The formula to calculate the fundamental frequency of a vibrating string is:

• \[ f = \frac{v}{2L} \]

where \(v\) is the wave speed and \(L\) is the length of the string. For a given tension, the speed \(v\) remains constant, so changing the length \(L\) alters the frequency. In this exercise, different lengths (49 cm and 50 cm) produced different fundamental frequencies, affecting the beat frequency with the tuning fork.