A stretched string is fixed at both its ends. Three possible wavelengths of stationary wave patterns that can be set up in the string are 90 cm, 60 cm and 45 cm. the length of the string may be

Options

A. 80 cm
B. 120 cm
C. 90 cm
D. 45 cm

Answer

To find the length of the string that can support stationary wave patterns with given wavelengths, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the relationship between wavelength and string length**:
The length of a string fixed at both ends can support standing waves that are integral multiples of half the wavelength. The relationship can be expressed as:
\[ L = n \frac{\lambda}{2} \]
where \( L \) is the length of the string, \( \lambda \) is the wavelength, and \( n \) is a positive integer (1, 2, 3,...).

2. **Setting up equations for each wavelength**:
Given the wavelengths \( \lambda_1 = 90 \, \text{cm} \), \( \lambda_2 = 60 \, \text{cm} \), and \( \lambda_3 = 45 \, \text{cm} \), we can set up equations for the length of the string:
- For \( \lambda_1 = 90 \, \text{cm} \):
\[ L = n_1 \frac{90}{2} = 45 n_1 \quad \text{(Equation 1)} \]
- For \( \lambda_2 = 60 \, \text{cm} \):
\[ L = n_2 \frac{60}{2} = 30 n_2 \quad \text{(Equation 2)} \]
- For \( \lambda_3 = 45 \, \text{cm} \):
\[ L = n_3 \frac{45}{2} = 22.5 n_3 \quad \text{(Equation 3)} \]

3. **Equating the lengths from different equations**:
Since all equations represent the same length \( L \), we can equate the expressions from Equation 1 and Equation 2:
\[ 45 n_1 = 30 n_2 \]
Rearranging gives:
\[ n_2 = \frac{45}{30} n_1 = \frac{3}{2} n_1 \quad \text{(Equation 4)} \]

4. **Equating lengths from Equation 2 and Equation 3**:
Now, equate Equation 2 and Equation 3:
\[ 30 n_2 = 22.5 n_3 \]
Rearranging gives:
\[ n_3 = \frac{30}{22.5} n_2 = \frac{4}{3} n_2 \quad \text{(Equation 5)} \]

5. **Substituting Equation 4 into Equation 5**:
Substitute \( n_2 = \frac{3}{2} n_1 \) into Equation 5:
\[ n_3 = \frac{4}{3} \left(\frac{3}{2} n_1\right) = 2 n_1 \]

6. **Finding the integer values of \( n_1 \)**:
Since \( n_1 \), \( n_2 \), and \( n_3 \) must be integers, we can set \( n_1 = 2 \) (the smallest integer that satisfies all equations):
- Then, \( n_2 = \frac{3}{2} \times 2 = 3 \)
- And \( n_3 = 2 \times 2 = 4 \)

7. **Calculating the length of the string**:
Now, substitute \( n_1 = 2 \) back into Equation 1 to find the length:
\[ L = 45 n_1 = 45 \times 2 = 90 \, \text{cm} \]

### Final Answer:
The length of the string is \( \boxed{90 \, \text{cm}} \).