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 analyze the vibrations of the two magnets in the Earth's magnetic field when they are aligned with the same polarity and when they are aligned with opposite polarity. Let's break down the solution step by step.

### Step 1: Understand the Given Information
- When the same poles (like poles) are together, the frequency of vibrations is 12 vibrations per minute.
- When opposite poles (unlike poles) are together, the frequency of vibrations is 4 vibrations per minute.

### Step 2: Convert Frequencies to Time Periods
The time period (T) is the reciprocal of the frequency (f). We can calculate the time periods for both cases.

1. **For the same poles:**
\[ f_1 = 12 \text{ vib/min} = \frac{12}{60} \text{ vib/sec} = 0.2 \text{ vib/sec} \]
\[ T_1 = \frac{1}{f_1} = \frac{1}{0.2} = 5 \text{ seconds} \]

2. **For the opposite poles:**
\[ f_2 = 4 \text{ vib/min} = \frac{4}{60} \text{ vib/sec} = \frac{1}{15} \text{ vib/sec} \]
\[ T_2 = \frac{1}{f_2} = \frac{1}{\frac{1}{15}} = 15 \text{ seconds} \]

### Step 3: Relate Time Periods to Magnetic Moments
The time periods for the two configurations can be related to the magnetic moments (m1 and m2) of the magnets using the following formulas:
- For the same poles:
\[ T_1 = 2\pi \sqrt{\frac{I_1 + I_2}{(m_1 + m_2)B_H}} \]
- For the opposite poles:
\[ T_2 = 2\pi \sqrt{\frac{I_1 + I_2}{(m_1 - m_2)B_H}} \]

### Step 4: Formulate the Ratio of Magnetic Moments
From the above equations, we can derive the ratio of the magnetic moments:
\[ \frac{T_1^2}{T_2^2} = \frac{m_1 - m_2}{m_1 + m_2} \]

### Step 5: Substitute Values
Substituting the values of \(T_1\) and \(T_2\):
\[ \frac{5^2}{15^2} = \frac{m_1 - m_2}{m_1 + m_2} \]
Calculating the squares:
\[ \frac{25}{225} = \frac{m_1 - m_2}{m_1 + m_2} \]
This simplifies to:
\[ \frac{1}{9} = \frac{m_1 - m_2}{m_1 + m_2} \]

### Step 6: Cross Multiply to Find the Ratio
Cross-multiplying gives:
\[ m_1 - m_2 = \frac{1}{9}(m_1 + m_2) \]
Multiplying through by 9:
\[ 9(m_1 - m_2) = m_1 + m_2 \]
Rearranging gives:
\[ 9m_1 - 9m_2 = m_1 + m_2 \]
\[ 8m_1 = 10m_2 \]
Thus, the ratio of the magnetic moments is:
\[ \frac{m_1}{m_2} = \frac{10}{8} = \frac{5}{4} \]

### Final Answer
The ratio of the magnetic moments \(m_1 : m_2\) is \(5 : 4\).

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