In a resonance tube experiment, the first two resonance are observed at length 10.5 cm and 29.5 cm. The third resonance is observed at the length …(cm)

Options

A. (a)`47.5 `
B. (b)`58.5`
C. (c)`48.5`
D. (d)`82.8`

Explanation

To find the length of the third resonance in a resonance tube experiment, we can follow these steps:

### Step 1: Understand the relationship between the resonance lengths
In a resonance tube experiment, the lengths at which resonance occurs correspond to the odd multiples of half wavelengths. The first resonance length \( L_1 \) and the second resonance length \( L_2 \) can be expressed as:
- \( L_1 = \frac{\lambda}{4} \) (first resonance)
- \( L_2 = \frac{3\lambda}{4} \) (second resonance)

### Step 2: Calculate the difference between the first two resonance lengths
The difference between the second and the first resonance lengths gives us the length of half a wavelength:
\[ L_2 - L_1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \]

Given:
- \( L_1 = 10.5 \, \text{cm} \)
- \( L_2 = 29.5 \, \text{cm} \)

Calculating the difference:
\[ L_2 - L_1 = 29.5 \, \text{cm} - 10.5 \, \text{cm} = 19.0 \, \text{cm} \]

### Step 3: Determine the wavelength
From the difference calculated, we can find the wavelength:
\[ \frac{\lambda}{2} = 19.0 \, \text{cm} \implies \lambda = 38.0 \, \text{cm} \]

### Step 4: Find the third resonance length
The third resonance occurs at:
\[ L_3 = L_1 + \lambda = 10.5 \, \text{cm} + 38.0 \, \text{cm} = 48.5 \, \text{cm} \]

### Conclusion
Thus, the length of the third resonance is:
\[ \boxed{48.5 \, \text{cm}} \]

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