In an experiment, a string is vibrating making 4 loops when 5 g is placed on the pan. Calculate the mass to be placed on the pan so as to make the string vibrate in 6 loops.

Answer

To solve the problem, we need to determine the mass that must be placed on the pan to make the string vibrate in 6 loops, given that it vibrates in 4 loops with a mass of 5 grams. We will use the relationship between the number of loops (harmonics) and the tension in the string.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Initial mass \( m_1 = 5 \) grams.
- Initial number of loops (harmonics) \( n_1 = 4 \).

2. **Determine the Tension:**
- The tension \( T_1 \) in the string when the mass \( m_1 \) is placed is given by:
\[ T_1 = m_1 \cdot g \]
- Here, \( g \) is the acceleration due to gravity, but since we are working in grams, we can keep \( g \) as a constant factor.

3. **Set Up the Relationship:**
- The relationship between the number of loops and tension can be expressed as:
\[ \frac{n_1}{n_2} = \sqrt{\frac{T_2}{T_1}} \]
- Where \( n_2 = 6 \) (the number of loops we want to achieve) and \( T_2 \) is the tension when the new mass \( m_2 \) is placed.

4. **Express Tension in Terms of Mass:**
- The tension for the new mass \( m_2 \) can be expressed as:
\[ T_2 = m_2 \cdot g \]

5. **Substituting Values:**
- Substitute \( T_1 \) and \( T_2 \) into the relationship:
\[ \frac{4}{6} = \sqrt{\frac{m_2 \cdot g}{5 \cdot g}} \]
- The \( g \) cancels out:
\[ \frac{4}{6} = \sqrt{\frac{m_2}{5}} \]

6. **Square Both Sides:**
- Squaring both sides gives:
\[ \left(\frac{4}{6}\right)^2 = \frac{m_2}{5} \]
- Simplifying:
\[ \frac{16}{36} = \frac{m_2}{5} \]

7. **Cross-Multiply to Solve for \( m_2 \):**
- Cross-multiplying gives:
\[ 16 \cdot 5 = 36 \cdot m_2 \]
- Thus:
\[ 80 = 36 \cdot m_2 \]

8. **Solve for \( m_2 \):**
- Finally, divide both sides by 36:
\[ m_2 = \frac{80}{36} \approx 2.22 \text{ grams} \]

### Final Answer:
The mass to be placed on the pan to make the string vibrate in 6 loops is approximately **2.22 grams**.