Questions & Answers

Question

Answers

A. 0.5 m

B. 1.0 m

C. 1.5 m

D. 2.0 m

Answer
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The formula for the length of the pipe is:

$L = \dfrac{nv}{2 \nu}$

Where n= 1 for fundamental frequency and n= 3 for second overtone.

In an open pipe, for fundamental frequency, to produce standing wave we require that the length of the pipe should be twice the wavelength of the fundamental mode or:

$\dfrac{\lambda}{2} = L$

The relation with frequency is:

$L = \dfrac{v}{2 \nu}$

where v is the velocity of sound in our case.

This is so because frequency and wavelength are related as:

$v=\nu \lambda$

For the second overtone, the wavelength is thrice of fundamental wavelength. Therefore, we replace $\lambda$ by $3 \times \lambda$

Keeping this in the formula for length and keeping the velocity of sound 333 m/s (given) in place of v:

$L = \dfrac{999 m/s}{2 \times 333 Hz}$

$L = \dfrac{ 3}{2}$m

L= 1.5 m

Therefore, the correct option is 1.5 m.

Standing waves result as a superposition of incoming waves and outgoing waves such that their phases become opposite. The best method to remember standing wave condition is to think of a rope with its one end tied. When we create a wave in such rope, at the tied end the wave reverses back and comes to us. Therefore, the entire rope seems to be making simple up and down motion.

The unit of frequency Hz is also $s^{-1}$ that is how we got cancellation of this in numerator and denominator and only m remained. Also the frequency and wavelength relation might get confusing. To sort it out, refer to the diagram; 3 times the new wavelength is twice the pipe length.