Problem 2831 - JEE Physics

Question

An air column in a pipe, which is closed at one end will be in resonance with a vibrating tuning fork of frequency \(264 \mathrm{~Hz}\), What is the length of the column if it is in \(\mathrm{cm} ?\) (speed of sound in \(\operatorname{air}=330 \mathrm{~m} / \mathrm{s}\) ) (A) \(62.50\) (B) \(15.62\) (C) 125 (D) \(93.75\)

Step-by-Step Solution

Answer

The length of the air column needed for resonance with the tuning fork is approximately 31.25 cm. The correct answer is not given in the options provided.

Step 1: Recall the fundamental resonance mode for a closed pipe

: The fundamental mode of resonance (n=1) for a closed pipe is given by the formula: \[f = \frac{2n - 1}{4L}v\] Where \(f\) is the frequency, \(L\) is the length of the air column, and \(v\) is the speed of sound in air.

Step 2: Substitute the given values into the formula

: We are given the frequency \(f = 264 \mathrm{~Hz}\) and the speed of sound in air \(v = 330 \mathrm{~m} / \mathrm{s}\). To solve for the length of the air column (L), we will substitute these values into the formula and solve for L. The given resonance mode for the closed pipe is the fundamental mode, n = 1: \[\frac{2(1) - 1}{4L} \times 330 = 264\]

Step 3: Solve for the air column length L

: Now, we need to solve the equation for L: \[\frac{1}{4L} \times 330 = 264\] To get L, we first multiply each side of the equation by 4L: \[330 = 264 \times 4L\] Now, divide each side by 264: \[L = \frac{330}{264 \times 4}\]

Step 4: Calculate the air column length and convert to cm

: Calculate the value of L: \[L = \frac{330}{1056} \approx 0.3125 \mathrm{~m}\] To convert this length to centimeters, multiply by 100: \[L = 0.3125 \times 100 \approx 31.25 \mathrm{~cm}\] The length of the air column needed for resonance with the tuning fork is approximately 31.25 cm. The closest answer in the options is (B) \(15.62\). However, this answer is not correct and thus the correct answer is not given in the options provided. The correct answer should be 31.25 cm.

Key Concepts

Fundamental Mode of Resonance

In the physics of sound, resonance occurs when an object vibrates at a specific frequency and causes another nearby object to vibrate. For closed pipes, the fundamental mode of resonance (also known as the first harmonic) happens when the standing wave pattern inside the pipe has a node at the closed end and an antinode at the open end.
In simpler terms, this is the lowest frequency at which the pipe will naturally resonate. The formula to determine this frequency for a closed pipe is:

• \( f = \frac{2n - 1}{4L}v \)

Where:

• \( f \) is the frequency,
• \( L \) is the length of the air column,
• \( v \) is the speed of sound in the medium,
• \( n \) is the harmonic number. For the fundamental mode, \( n = 1 \).

This means only a quarter of a wavelength fits inside the closed pipe.
Understanding this concept is crucial for various applications, including musical instruments and acoustics engineering.

Frequency of Tuning Fork

A tuning fork produces a constant pitch, helping musicians tune their instruments by creating a specific frequency sound wave. In this scenario, the frequency of the tuning fork is given as 264 Hz.
The frequency of a sound relates to how many wave crests pass a point per second. Higher frequencies mean a higher pitch.
When the fork vibrates, it creates sound waves that travel through the air column in the pipe, setting it into resonance. The resonating length of the air column needs to match the wave pattern created by the tuning fork's frequency.
This scenario showcases the interaction between the mechanical vibrations produced by the fork and the resonating air column, highlighting how specific frequencies affect different acoustic systems.

Speed of Sound in Air

Sound travels through air at a specific speed, determined by factors like temperature, humidity, and air composition. Here, the speed of sound is given as 330 m/s, a common value used in physics exercises under standard conditions.
Sound speed affects how quickly waves move through a medium. In this case, the speed of sound in air is crucial for determining the air column length that will resonate with the tuning fork's frequency.
The speed of sound can change in different environments, but for the sake of this problem, we assume the standard value. Understanding how sound speed interacts with wave patterns helps explain phenomena in meteorology and engineering.

Air Column Length Calculation

The goal of this exercise is to determine the length of an air column in a closed pipe that resonates with a tuning fork of known frequency. Calculating this length involves manipulating the fundamental mode's formula, substituting known values, and solving for the unknown.
We know:

• Frequency \( f = 264 \) Hz,
• Speed of sound \( v = 330 \) m/s,
• \( n = 1 \) for the fundamental mode.

Given the equation \( f = \frac{2n - 1}{4L}v \), substituting the values yields this step:

• \( \frac{1}{4L} \times 330 = 264 \)

Solving for \( L \) involves:

• \( L = \frac{330}{264 \times 4} = \frac{330}{1056} \)

This calculation gives \( L = 0.3125 \) meters, or 31.25 cm when converted.
This result allows us to understand how physical characteristics and known measures interact, forming a basis for more complex acoustic designs and systems.