in an experiment it was found that string vibrates in n loops when a mass M is placed on the pan. What mass should be placed on the pan to make it vibrate in 2n loops with same frequency ? ( neglect the mass of pan )

Options

A. 2m
B. `m/4`
C. 4M
D. `M/2`

Answer

To solve the problem, we need to determine the mass that should be placed on the pan to make the string vibrate in 2n loops while maintaining the same frequency. Let's break down the solution step by step:

### Step-by-Step Solution:

1. **Understanding the relationship between loops and wavelength**:
- The length of the string (L) is related to the number of loops (n) and the wavelength (λ) by the formula:
\[ L = n \cdot \frac{\lambda}{2} \]
- This means that for n loops, the wavelength is:
\[ \lambda = \frac{2L}{n} \]

2. **Determining the new wavelength for 2n loops**:
- When the string vibrates in 2n loops, the new wavelength (λ') becomes:
\[ \lambda' = \frac{2L}{2n} = \frac{L}{n} \]

3. **Using the frequency formula**:
- The frequency (f) of the wave is given by:
\[ f = \frac{V}{\lambda} \]
- Since the frequency must remain constant, we have:
\[ f = \frac{V}{\lambda} = \frac{V'}{\lambda'} \]
- This implies:
\[ V = f \cdot \lambda \quad \text{and} \quad V' = f \cdot \lambda' \]

4. **Relating the velocities**:
- Since λ' is half of λ (λ' = λ/2), we can express the velocities as:
\[ V' = f \cdot \frac{L}{n} \]
- Thus, if λ is halved, the new velocity (V') must also change.

5. **Understanding the relationship between velocity and tension**:
- The velocity of a wave on a string is related to the tension (T) and the mass per unit length (μ) of the string:
\[ V = \sqrt{\frac{T}{\mu}} \]
- If the velocity is halved (V' = V/2), then:
\[ V' = \sqrt{\frac{T'}{\mu}} \]
- This means:
\[ \left(\frac{V}{2}\right)^2 = \frac{T'}{\mu} \]
- Therefore, if V is halved, the new tension (T') becomes:
\[ T' = \frac{T}{4} \]

6. **Relating tension to mass**:
- The tension in the string is directly proportional to the mass (M) placed on the pan:
\[ T \propto M \]
- Therefore, if the new tension is T' = T/4, then the new mass (M') that should be placed on the pan is:
\[ M' = \frac{M}{4} \]

### Final Answer:
To make the string vibrate in 2n loops with the same frequency, the mass that should be placed on the pan is:
\[ \boxed{\frac{M}{4}} \]