An air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length of the column is $50\,cm$. The next larger length of the column resonating with the same tuning fork is :
- 100 cm
- 150 cm
- 200 cm
- 66.7 cm
The Correct Option is B
Approach Solution - 1
Resonating Length for a Closed Organ Pipe
The first minimum resonating length for a closed organ pipe is given by:
\(\frac{\lambda}{4} = 50 \, \text{cm}\)
Where: - \( \lambda \) is the wavelength of the sound wave in the pipe.
\(\therefore\) The next larger length of the air column is:
\(\frac{3 \lambda}{4} = 150 \, \text{cm}\)
Conclusion:
The next resonating length of the air column is 150 cm.
Approach Solution -2
Concept: For an air column closed at one end and open at the other, resonance occurs at odd multiples of the quarter wavelength.
So, the resonant lengths are:
\(L_1 = \frac{\lambda}{4},\quad L_2 = \frac{3\lambda}{4},\quad L_3 = \frac{5\lambda}{4},\quad \text{and so on.}\)
Given:
- First resonant length \(L_1 = 50\, \text{cm}\)
Next resonant length:
\(L_2 = \frac{3\lambda}{4} = 3 \times \frac{\lambda}{4} = 3 \times L_1 = 3 \times 50 = 150\, \text{cm}\)
Answer: \(\boxed{150\, \text{cm}}\)
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Concepts Used:
Waves
Waves are a disturbance through which the energy travels from one point to another. Most acquainted are surface waves that tour on the water, but sound, mild, and the movement of subatomic particles all exhibit wavelike properties. inside the most effective waves, the disturbance oscillates periodically (see periodic movement) with a set frequency and wavelength.
Types of Waves:
Transverse Waves -
Waves in which the medium moves at right angles to the direction of the wave.
Examples of transverse waves:
- Water waves (ripples of gravity waves, not sound through water)
- Light waves
- S-wave earthquake waves
- Stringed instruments
- Torsion wave
The high point of a transverse wave is a crest. The low part is a trough.
Longitudinal Wave -
A longitudinal wave has the movement of the particles in the medium in the same dimension as the direction of movement of the wave.
Examples of longitudinal waves:
- Sound waves
- P-type earthquake waves
- Compression wave