Two magnets held together in earth's magnetic field when the same polarity together causes 12 vib/min and when opposite poles 4 vib/min. What is the ratio of magnetic moments?

Answer

To solve the problem, we need to determine the ratio of the magnetic moments of two magnets based on their vibrations in the Earth's magnetic field when placed with the same and opposite polarities.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two magnets that vibrate at different frequencies depending on whether they are aligned with the same polarity or opposite polarities. We need to find the ratio of their magnetic moments.

2. **Identify Frequencies**:
- When the same polarities are together, the frequency of vibration (v1) is 12 vibrations per minute.
- When opposite poles are together, the frequency of vibration (v2) is 4 vibrations per minute.

3. **Convert Frequencies to Time Periods**:
The time period (T) is the reciprocal of frequency (f). Since the frequency is given in vibrations per minute, we convert it to seconds:
- For same polarity (v1 = 12 vib/min):
\[ T_1 = \frac{60 \text{ seconds}}{12 \text{ vib/min}} = 5 \text{ seconds} \]
- For opposite polarity (v2 = 4 vib/min):
\[ T_2 = \frac{60 \text{ seconds}}{4 \text{ vib/min}} = 15 \text{ seconds} \]

4. **Use the Formula for Magnetic Moments**:
The ratio of the magnetic moments (M1/M2) can be calculated using the formula:
\[ \frac{M_1}{M_2} = \frac{T_2^2 + T_1^2}{T_2^2 - T_1^2} \]

5. **Substituting the Values**:
- Calculate \(T_1^2\) and \(T_2^2\):
\[ T_1^2 = 5^2 = 25 \]
\[ T_2^2 = 15^2 = 225 \]
- Substitute these values into the formula:
\[ \frac{M_1}{M_2} = \frac{225 + 25}{225 - 25} = \frac{250}{200} = \frac{5}{4} \]

6. **Final Result**:
The ratio of the magnetic moments of the two magnets is:
\[ \frac{M_1}{M_2} = \frac{5}{4} \]