Question

A vibrating source over the mouth of a closed organ pipe is in unison at temperature t .The beats produced when temperature rises by t are N. Find the frequency of the vibrating source:
$\begin{align}
  & a)\dfrac{5N}{3t} \\
 & b)\dfrac{2NT}{t} \\
 & c)\dfrac{2N(t-T)}{t} \\
 & d)\dfrac{N(t+T)}{t} \\
\end{align}$

Answer 1

Hint: Find out the frequency in a closed organ pipe. As the temperature is increased, the velocity also changes. So, the new frequency is calculated using the new velocity of the source. As the difference between both the frequencies is given, we can find the frequency in the known terms.
Formulas used:
$\begin{align}
  & f=\dfrac{V}{4l} \\
 & beats={{f}^{1}}-f \\
\end{align}$

Complete answer:
The frequency of the closed organ pipe is given as,
$f=\dfrac{V}{4l}$
This frequency is the frequency at temperature t and speed v.
Now, as the temperature is increased by t,
The new frequency will be equal to,
${{f}^{1}}=\dfrac{V(1+0.6t)}{4l}$
This frequency is the frequency at a new temperature.
Now, the difference between the new frequencies is given as n.
\[\begin{align}
  & N={{f}^{1}}-f \\
 & \\
\end{align}\]
Dividing the two frequencies, we get,
$\begin{align}
  & {{f}^{1}}=f(1+0.6t) \\
 & {{f}^{1}}-f=0.6ft \\
 & N=0.6ft \\
 & f=\dfrac{N}{0.6t} \\
 & f=\dfrac{5N}{3t} \\
\end{align}$

Therefore, the correct option is option a.

Additional information:
Closed organ pipe is the organ pipe in which one end is opened and another end is closed. Best examples are bottles, whistles etc. In the first mode of vibration in the closed organ pipe, an antinode is formed at the open end and a node is formed at the closed end. A closed cylindrical air Column will produce resonant standing waves at fundamental frequencies and at odd harmonics. The closed end is constrained to be a node of the wave and the open end is of course an antinode. The air at the end of a closed pipe is not free to vibrate , therefore, a node is formed at the closed end. In the open end, the air is free to vibrate at maximum amplitude which makes an antinode to be firmed at the open end.

Note:
The frequency of the closed pipe is velocity divided by four times the length of the tube. But, in the case of an open pipe, the frequency of the open pipe is velocity divided by two times the length of the pipe. Therefore, take care of the formulas used in the open pipe frequency and the closed pipe frequency.