Three strings of equal lengths but stretched with different tensions are inade to vibrate. If their masses per unit length are in the ratio 1:2:3 and the frequencies are the same, calculate the ratio of the tensions.

Answer

To solve the problem, we need to find the ratio of tensions in three strings that have the same length but different tensions and different masses per unit length, while their frequencies are the same.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- Let the tensions in the three strings be \( T_1, T_2, T_3 \).
- Let the masses per unit length of the three strings be \( m_1, m_2, m_3 \).
- The ratio of the masses per unit length is given as \( m_1 : m_2 : m_3 = 1 : 2 : 3 \).

2. **Using the Frequency Formula:**
- The frequency \( f \) of a vibrating string is given by the formula:
\[ f = \frac{1}{L} \sqrt{\frac{T}{m}} \]
- Since the lengths of the strings are the same, we can simplify this to:
\[ f = \sqrt{\frac{T}{m}} \]

3. **Setting Up the Equations:**
- Since the frequencies of all three strings are the same, we can write:
\[ \sqrt{\frac{T_1}{m_1}} = \sqrt{\frac{T_2}{m_2}} = \sqrt{\frac{T_3}{m_3}} = k \]
- Where \( k \) is a constant.

4. **Squaring the Equations:**
- Squaring the equations gives us:
\[ \frac{T_1}{m_1} = \frac{T_2}{m_2} = \frac{T_3}{m_3} = C \]
- Where \( C \) is another constant.

5. **Expressing Tensions in Terms of Masses:**
- From the above equations, we can express the tensions in terms of the masses:
\[ T_1 = C \cdot m_1, \quad T_2 = C \cdot m_2, \quad T_3 = C \cdot m_3 \]

6. **Finding the Ratio of Tensions:**
- The ratio of the tensions can now be expressed as:
\[ T_1 : T_2 : T_3 = m_1 : m_2 : m_3 \]
- Substituting the given ratio of masses:
\[ T_1 : T_2 : T_3 = 1 : 2 : 3 \]

### Final Answer:
The ratio of the tensions in the three strings is \( T_1 : T_2 : T_3 = 1 : 2 : 3 \).

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