Question: 45. A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to:
(1) \frac{f}{2}
(2) f
(3) \frac{3 f}{2}
(4) 2 f
Answer: Option (2)
Explanation:
Let the original length of the pipe be L.
Since the pipe is open at both ends,
the fundamental frequency of the air column is given by
f = \frac{v}{2L}, where v
is the speed of sound in air.
When the pipe is dipped vertically in water up to half of its length,
the lower half of the pipe gets filled with water.
The length of the air column is now \frac{L}{2}.
The water surface behaves like a closed end for the air column,
while the upper end of the pipe remains open.
Thus, the air column now acts as a pipe closed at one end with length \frac{L}{2}.
The fundamental frequency of a closed pipe is given by
f' = \frac{v}{4l}, where l is the length of the air column.
Substituting l = \frac{L}{2},
we get f' = \frac{v}{4 \times \frac{L}{2}} = \frac{v}{2L}.
Since the original frequency was f = \frac{v}{2L},
the new fundamental frequency is equal to f.
Therefore, the correct option is Option (2).