Organ Pipe Harmonics Analysis
This problem involves comparing the frequencies of harmonics in a closed organ pipe and an open organ pipe.
Frequencies of Organ Pipes
- Closed Organ Pipe: The allowed frequencies (harmonics) are given by the formula $f_n = \frac{nv}{4L_c}$, where $n$ can only be odd integers (1, 3, 5, ...), $v$ is the speed of sound, and $L_c$ is the length of the pipe.
- Open Organ Pipe: The allowed frequencies are given by the formula $f'_m = \frac{mv}{2L_o}$, where $m$ can be any positive integer (1, 2, 3, ...), $v$ is the speed of sound, and $L_o$ is the length of the pipe.
Harmonic Condition
The problem states that the fifth harmonic of the closed pipe ($n=5$) is in unison (equal frequency) with the first harmonic of the open pipe ($m=1$).
Frequency of the 5th harmonic of the closed pipe: $f_5 = \frac{5v}{4L_c}$.
Frequency of the 1st harmonic of the open pipe: $f'_1 = \frac{1 \times v}{2L_o} = \frac{v}{2L_o}$.
Equating Frequencies
Since the frequencies are equal:
$ \frac{5v}{4L_c} = \frac{v}{2L_o} $
Calculating Length Ratio
We can cancel $v$ from both sides and rearrange the equation to find the ratio $\frac{L_c}{L_o}$:
$ \frac{5}{4L_c} = \frac{1}{2L_o} $
$ \frac{L_c}{L_o} = \frac{5}{4} \times \frac{2}{1} $
$ \frac{L_c}{L_o} = \frac{10}{4} = \frac{5}{2} $
Determining the Value of x
The problem gives the ratio of the lengths as $\frac{L_c}{L_o} = \frac{5}{x}$.
Comparing this with our calculated ratio:
$ \frac{5}{x} = \frac{5}{2} $
Therefore, the value of $x$ is 2.
Similar Questions
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List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
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Important Questions from Mechanics
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
