Two magnets are held together in a vibration magnetometer and are allowed to oscillate in the earth's magnetic field with like poles togather, 12 oscillations per minute are made but for unlike poles togather only 4 oscillations per minute are axecuted. The ratio of their magnetic miments is

Options

A. `3:1`
B. `1:3`
C. `3:5`
D. `5:4`

Explanation

To solve the problem, we need to find the ratio of the magnetic moments of two magnets based on their oscillation frequencies when held together in a vibration magnetometer. Here's a step-by-step solution:

### Step 1: Understand the given data
- Frequency of oscillation with like poles together (f1) = 12 oscillations per minute
- Frequency of oscillation with unlike poles together (f2) = 4 oscillations per minute

### Step 2: Convert frequencies to time periods
The time period (T) is the reciprocal of frequency (f). Since the frequency is given in oscillations per minute, we will convert it to seconds.

- For like poles together:
\[ T_1 = \frac{60}{f_1} = \frac{60}{12} = 5 \text{ seconds} \]

- For unlike poles together:
\[ T_2 = \frac{60}{f_2} = \frac{60}{4} = 15 \text{ seconds} \]

### Step 3: Use the formula for the ratio of magnetic moments
The relationship between the time periods and the magnetic moments can be expressed as:
\[ \frac{M_1}{M_2} = \frac{T_1^2 + T_2^2}{T_2^2 - T_1^2} \]

### Step 4: Calculate \(T_1^2\) and \(T_2^2\)
- \(T_1^2 = 5^2 = 25\)
- \(T_2^2 = 15^2 = 225\)

### Step 5: Substitute values into the formula
Now, substitute \(T_1^2\) and \(T_2^2\) into the ratio formula:
\[ \frac{M_1}{M_2} = \frac{T_1^2 + T_2^2}{T_2^2 - T_1^2} = \frac{25 + 225}{225 - 25} = \frac{250}{200} \]

### Step 6: Simplify the ratio
\[ \frac{M_1}{M_2} = \frac{250}{200} = \frac{5}{4} \]

### Final Answer
The ratio of the magnetic moments of the two magnets is:
\[ \frac{M_1}{M_2} = \frac{5}{4} \]