Question

A tuning fork of frequency 340 Hz resonates in the fundamental mode with an air column of length 125 cm in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is ____ cm.
(Velocity of sound in air is 340 ms^–1)

Answer 1

Correct Answer: 50

To solve this problem, we start by understanding that resonance occurs when the length of the air column corresponds to a specific mode of vibration. In a tube closed at one end, the lowest resonant frequency (fundamental frequency) occurs when the length of the air column is one-fourth of the wavelength (λ/4). Given:

• Frequency (f) = 340 Hz
• Velocity of sound (v) = 340 m/s
• Length of air column (L) = 125 cm = 1.25 m

For the fundamental mode:

v = fλ ⇒ λ = v/f = 340/340 = 1 m

At fundamental resonance, L = λ/4 = 1/4 m = 0.25 m

The air column is initially resonating in the fundamental mode at 1.25 m, but the actual resonant length for the fundamental frequency is 0.25 m. This implies that the next resonance (first overtone) occurs when the length meets the next odd multiple of λ/4, i.e., 3λ/4. Therefore, L_next = 3 × 0.25 m = 0.75 m.

The reduction in length required for next resonance when the air column is resonating again is given by:

Reduction = Initial Length - Next Resonant Length = 1.25 m - 0.75 m = 0.50 m = 50 cm

Thus, the minimum height of water required for the next resonance is 50 cm. The calculated result of 50 cm aligns perfectly with the expected range of 50, 50 cm.