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
- 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 _________ $\text{Hz}$.
- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
- A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x=0 at t=0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?
A particle is subjected to two simple harmonic motions as :
$x_1 = \sqrt{7} \sin 5t \text{ cm}$ and $x_2 = 2\sqrt{7} \sin \left(5t + \frac{\pi}{3}\right) \text{ cm}$
where x is displacement and t is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \text{ ms}^{-2}$. The value of x is :- 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 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 velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}~\text{s}$, the value of $x$ is ________.
Two simple harmonic motions, as shown below, are at right angles. They are combined to form lissajous figures.
$x(t) = A \sin (at + \delta)$
$y(t) = B \sin (bt)$
Identify the correct match below :
The motion of a mass on a spring, with spring constant K is as shown in figure.
The equation of motion is given by $x(t) = A\sin\omega t+B\cos\omega t$ with $\omega=\sqrt{\frac{K}{m}}$.
Suppose that at time $t = 0$, the position of mass is $x(0)$ and velocity $v(0)$, then its displacement can also be represented as $x(t) = C\cos(\omega t-\phi)$, where $C$ and $\phi$ are
- $T_0$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, the modified time period is
Important Questions from Oscillations and Waves
- 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 _________ $\text{Hz}$.
- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
- A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x=0 at t=0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?
A particle is subjected to two simple harmonic motions as :
$x_1 = \sqrt{7} \sin 5t \text{ cm}$ and $x_2 = 2\sqrt{7} \sin \left(5t + \frac{\pi}{3}\right) \text{ cm}$
where x is displacement and t is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \text{ ms}^{-2}$. The value of x is :- The equation of a transverse wave is $y = y_0 \sin 2\pi(ft - \frac{x}{\lambda})$. If the maximum particle velocity be four times that of wave velocity then
