Question

Equal volumes of water are added to three cylindrical jars A, B and C of the same height and radii \(r_A\), \(r_B\) and \(r_C\) respectively with \(r_A<r_B<r_C\). If you blow across the mouth of these jars, which tube will produce the shrillest note?

• A. A
• B. B
• C. C
• D. All will produce the notes of same shrillness

Hint:

For resonance in air columns, remember: Short air column = High frequency = Shrill sound. Long air column = Low frequency = Grave sound.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question is about the sound produced by blowing over the top of three cylindrical jars with different radii but filled with the same volume of water. We need to find which jar produces the shrillest note (highest frequency).
Step 2: Key Formula or Approach:
The sound is produced by the resonance of the air column above the water. The pitch (frequency) of the sound produced by a vibrating air column in a closed pipe (like a jar) is inversely proportional to the length of the air column.
\[ f \propto \frac{1}{L} \] where \(f\) is the frequency and \(L\) is the length of the air column. A shriller note corresponds to a higher frequency.
Step 3: Detailed Explanation:
Let H be the total height of each cylindrical jar.
Let V be the equal volume of water added to each jar.
The volume of water in a cylinder is given by \(V = \text{base area} \times \text{height of water} = \pi r^2 h_w\).
Since V is the same for all jars, the height of the water (\(h_w\)) is \(h_w = \frac{V}{\pi r^2}\).
This means \(h_w \propto \frac{1}{r^2}\).
We are given the radii are in the order \(r_A<r_B<r_C\).
Therefore, the heights of the water will be in the reverse order: \(h_{wA}>h_{wB}>h_{wC}\).
The length of the vibrating air column (\(L\)) in each jar is the total height minus the water height: \(L = H - h_w\).
Since \(h_{wA}\) is the largest water height, the corresponding air column length \(L_A = H - h_{wA}\) will be the smallest.
So, the lengths of the air columns are in the order \(L_A<L_B<L_C\).
The frequency is inversely proportional to the length of the air column (\(f \propto 1/L\)).
Therefore, the frequencies will be in the order \(f_A>f_B>f_C\).
A higher frequency means a shriller note. Since \(f_A\) is the highest frequency, jar A will produce the shrillest note.
Step 4: Final Answer:
Jar A has the smallest radius, so for the same volume of water, it will have the highest water level, leaving the shortest air column. A shorter air column vibrates at a higher frequency, producing a shriller sound.