In an open organ pipe $\nu_3$ and $\nu_6$ are $3^{\text{rd}}$ and $6^{\text{th}}$ harmonic frequencies, respectively. If $\nu_6 - \nu_3 = 2200 \text{ Hz}$ then length of the pipe is _________ mm.
(Take velocity of sound in air is $330 \text{ m/s}$.)
Open Organ Pipe Harmonics Calculation
For an open organ pipe, the frequencies of the harmonics are given by the formula:
$ \nu_n = \frac{n v}{2L} $
where $n$ is the harmonic number ($n=1, 2, 3, \ldots$), $v$ is the velocity of sound, and $L$ is the length of the pipe.
Harmonic Frequency Analysis
- The 3rd harmonic frequency ($\nu_3$) is: $ \nu_3 = \frac{3v}{2L} $
- The 6th harmonic frequency ($\nu_6$) is: $ \nu_6 = \frac{6v}{2L} $
Calculating Pipe Length
We are given the difference between the 6th and 3rd harmonic frequencies:
$ \nu_6 - \nu_3 = 2200 \text{ Hz} $
Substituting the formulas for $\nu_6$ and $\nu_3$:
$ \frac{6v}{2L} - \frac{3v}{2L} = 2200 $
Simplify the equation:
$ \frac{(6-3)v}{2L} = 2200 $
$ \frac{3v}{2L} = 2200 $
Now, we solve for the length $L$. Rearranging the equation:
$ L = \frac{3v}{2 \times 2200} $
Given the velocity of sound $v = 330 \text{ m/s}$:
$ L = \frac{3 \times 330 \text{ m/s}}{4400 \text{ Hz}} $
$ L = \frac{990}{4400} \text{ m} $
$ L = \frac{99}{440} \text{ m} = \frac{9}{40} \text{ m} $
To convert the length from meters to millimeters, multiply by 1000:
$ L = \frac{9}{40} \times 1000 \text{ mm} $
$ L = 9 \times 25 \text{ mm} $
$ L = 225 \text{ mm} $
The length of the pipe is 225 mm.
Similar Questions
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
- The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of $176\text{ rad/s}$. The frequency of this simple harmonic oscillator is __________ $\text{Hz}$. $\left[ \text{take } \pi = \frac{22}{7} \right]$
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is 5 sec. The time required by it to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is _________ sec.
- Match List - I with List - II.
List - I List - II A. $\sin^{2} \omega t$ I. Periodic with time period $T=\frac{\pi}{\omega}$ but not simple harmonic motion (SHM) B. $\sin^{3} (2\omega t)$ II. Periodic with time period $T=\frac{2\pi}{\omega}$ but Not SHM C. $\sin (\omega t) + \cos(\pi \omega t)$ III. Periodic with time period $T=\frac{\pi}{\omega}$ and SHM D. $\cos \omega t + \cos 2\omega t$ IV. Non-periodic
Choose the correct answer from the options given below :
Important Questions from Oscillations and Waves
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
- The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of $176\text{ rad/s}$. The frequency of this simple harmonic oscillator is __________ $\text{Hz}$. $\left[ \text{take } \pi = \frac{22}{7} \right]$
