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 step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Problem
We have two bar magnets that can be aligned in two different ways: like poles together and opposite poles together. We need to find the ratio of their magnetic moments based on the time periods of oscillation in these two configurations.

### Step 2: Write Down the Given Information
- Time period when like poles are aligned (T1) = 6 seconds
- Time period when opposite poles are aligned (T2) = 12 seconds

### Step 3: Recall the Formula for Time Period
The time period (T) of a magnet suspended in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{MB}} \]
where:
- \(I\) = moment of inertia
- \(M\) = magnetic moment
- \(B\) = magnetic field strength

### Step 4: Analyze the Two Cases
1. **Like poles aligned**: The effective magnetic moment \(M\) is the sum of the two magnetic moments:
\[ M = M_1 + M_2 \]
Thus, the time period \(T_1\) can be expressed as:
\[ T_1 \propto \sqrt{\frac{I}{(M_1 + M_2)B}} \]

2. **Opposite poles aligned**: The effective magnetic moment \(M\) is the difference of the two magnetic moments:
\[ M = M_1 - M_2 \]
Thus, the time period \(T_2\) can be expressed as:
\[ T_2 \propto \sqrt{\frac{I}{(M_1 - M_2)B}} \]

### Step 5: Set Up the Ratios
From the above expressions, we can set up the following ratios based on the time periods:
\[ \frac{T_2}{T_1} = \sqrt{\frac{M_1 + M_2}{M_1 - M_2}} \]

### Step 6: Substitute the Values
Substituting the known values of \(T_1\) and \(T_2\):
\[ \frac{12}{6} = \sqrt{\frac{M_1 + M_2}{M_1 - M_2}} \]
This simplifies to:
\[ 2 = \sqrt{\frac{M_1 + M_2}{M_1 - M_2}} \]

### Step 7: Square Both Sides
Squaring both sides gives:
\[ 4 = \frac{M_1 + M_2}{M_1 - M_2} \]

### Step 8: Cross Multiply
Cross multiplying gives:
\[ 4(M_1 - M_2) = M_1 + M_2 \]
Expanding this yields:
\[ 4M_1 - 4M_2 = M_1 + M_2 \]

### Step 9: Rearrange the Equation
Rearranging the equation results in:
\[ 4M_1 - M_1 = 4M_2 + M_2 \]
This simplifies to:
\[ 3M_1 = 5M_2 \]

### Step 10: Find the Ratio
Dividing both sides by \(M_2\) gives:
\[ \frac{M_1}{M_2} = \frac{5}{3} \]

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

---