Chapter 17: Q40P (page 509)

Organ pipe $A$ , with both ends open, has a fundamental frequency of $300Hz$ . The third harmonic of organ piperole="math" localid="1661418890848" $B$ , with one end open, has the same frequency as the second harmonic of pipe $A$ . How long are (a) pipe $A$ and (b) pipe $B$ ?

Short Answer

Expert verified

The length of pipe A is $0.572 m$ .
The length of pipe B is $0.429 m$ .

Step by step solution

Given data

Fundamental frequency , $f_{0} = 300 Hz$ ( for pipe A)
Second harmonics of A = 3^rd harmonics of B

Determining the concept

Apply the formula for harmonics for a closed and open organ pipe.

The expression for the frequency is given by,

$f_{0} = \frac{n}{2 L_{A}} v$

Here, $f_{0}$ is frequency, $L$ is length and $v$ is the velocity.

(a) Determine the length of pipe A

Fundamental frequency is defined as,

$f_{0} = \frac{n}{2 L_{A}} v$

For fundamental frequency, $n = 1$

$f_{0} = \frac{v}{2 L_{A}}$

$\begin{matrix} L_{A} = \frac{v}{2 f_{0}} \\ = \frac{(343 m / s)}{2 \times 300 Hz} \\ = 0.572 m \end{matrix}$

Hence,the length of pipe A is $0.572 m$ .

(b) Determine the length of pipe B

Frequency of harmonics for one end open pipe is defined as,

$f_{n} = (n - \frac{1}{2}) (\frac{v}{2 L_{B}})$

$Forthirdharmonicsmeanssecondoverton, n = 2$

$\begin{matrix} f_{3} = (2 - \frac{1}{2}) (\frac{v}{2 L_{B}}) \\ = \frac{3 \times 343 m / s}{2 \times 2 \times L_{B}} \end{matrix}$

$f_{3} = \frac{257.25}{L_{B}} Hz$

Now, second harmonic of A = third harmonic of B

$2 f_{0} = f_{3}$

$2 (300 Hz) = \frac{257.25}{L_{B}} Hz-m$

$\begin{matrix} L_{B} = \frac{257.25m}{600} \\ = 0.429 m \end{matrix}$

Hence,the length of pipe B is $0.429 m$ .

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Organ pipe $A$ , with both ends open, has a fundamental frequency of $300Hz$ . The third harmonic of organ piperole="math" localid="1661418890848" $B$ , with one end open, has the same frequency as the second harmonic of pipe $A$ . How long are (a) pipe $A$ and (b) pipe $B$ ?

Short Answer

Step by step solution

Given data

Determining the concept

(a) Determine the length of pipe A

(b) Determine the length of pipe B

One App. One Place for Learning.

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Organ pipe A, with both ends open, has a fundamental frequency of300Hz. The third harmonic of organ piperole="math" localid="1661418890848" B, with one end open, has the same frequency as the second harmonic of pipeA. How long are (a) pipeAand (b) pipeB?

Short Answer

Step by step solution

Given data

Determining the concept

(a) Determine the length of pipe A

(b) Determine the length of pipe B

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Wave Optics

Circular Motion and Gravitation

Scientific Method Physics

Further Mechanics and Thermal Physics

Energy Physics

Study anywhere. Anytime. Across all devices.

Organ pipe $A$ , with both ends open, has a fundamental frequency of $300Hz$ . The third harmonic of organ piperole="math" localid="1661418890848" $B$ , with one end open, has the same frequency as the second harmonic of pipe $A$ . How long are (a) pipe $A$ and (b) pipe $B$ ?