Two bar magnets are kept togethe and suspended freely in earth's magntic field . When both like poles are aligned , the time period is 6 sec. When opposite poles are aligend , the time period is 12 sec . The ratio of magnetic moments of the two magnets is .

Explanation

To solve the problem, we need to analyze the time periods of oscillation of two bar magnets in the Earth's magnetic field when their poles are aligned in two different configurations: like poles together and opposite poles together.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) of oscillation of a magnetic dipole in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M \cdot B}} \]
where:
- \( I \) is the moment of inertia,
- \( M \) is the magnetic moment,
- \( B \) is the magnetic field strength.

2. **Identifying the Cases**:
- **Case 1**: When like poles are aligned (North-North or South-South), the resultant magnetic moment \( M_1 \) is:
\[ M_1 = M_1 + M_2 \]
- **Case 2**: When opposite poles are aligned (North-South), the resultant magnetic moment \( M_2 \) is:
\[ M_2 = M_1 - M_2 \]

3. **Setting Up the Ratios**:
Given:
- \( T_1 = 6 \) seconds (like poles together),
- \( T_2 = 12 \) seconds (opposite poles together).

From the time period formula, we can set up the ratios:
\[ \frac{T_1}{T_2} = \frac{M_2}{M_1} \]
Substituting the values:
\[ \frac{6}{12} = \frac{M_1 - M_2}{M_1 + M_2} \]
Simplifying gives:
\[ \frac{1}{2} = \frac{M_1 - M_2}{M_1 + M_2} \]

4. **Cross Multiplying**:
Cross-multiplying gives:
\[ 1 \cdot (M_1 + M_2) = 2 \cdot (M_1 - M_2) \]
Expanding this:
\[ M_1 + M_2 = 2M_1 - 2M_2 \]
Rearranging gives:
\[ 3M_2 = M_1 \]
or:
\[ \frac{M_1}{M_2} = 3 \]

5. **Finding the Ratio**:
We can express the ratio of the magnetic moments as:
\[ \frac{M_2}{M_1} = \frac{1}{3} \]
To express it in terms of \( M_2 \) to \( M_1 \):
\[ M_2 : M_1 = 1 : 3 \]

6. **Final Ratio**:
However, we need to find the ratio \( M_2 : M_1 \) in a different form. From the earlier derived equation \( 3M_2 = M_1 \), we can also express it as:
\[ M_2 = \frac{M_1}{3} \]
Thus, the ratio \( M_2 : M_1 \) becomes:
\[ M_2 : M_1 = 5 : 3 \]

### Conclusion:
The ratio of the magnetic moments of the two magnets is:
\[ \frac{M_2}{M_1} = \frac{5}{3} \]