Two magnets of same size and mass make respectively 10 and 15 oscillations per minute at certain place. The ratio of their magnetic moment is

Options

A. `4:9`
B. `9:4`
C. `2:3`
D. `3:2`

Answer

To find the ratio of the magnetic moments of two magnets based on their oscillation frequencies, we can follow these steps:

### Step 1: Understand the given data
We have two magnets:
- Magnet 1 (M1) makes 10 oscillations per minute.
- Magnet 2 (M2) makes 15 oscillations per minute.

### Step 2: Convert oscillations per minute to frequency
Frequency (f) is defined as the number of oscillations per second. Since there are 60 seconds in a minute, we can convert the oscillations per minute to oscillations per second (Hz):
- Frequency of M1, \( f_1 = \frac{10}{60} = \frac{1}{6} \) Hz
- Frequency of M2, \( f_2 = \frac{15}{60} = \frac{1}{4} \) Hz

### Step 3: Use the formula for frequency of oscillation
The frequency of oscillation of a magnet is given by the formula:
\[ f = \frac{1}{2\pi} \sqrt{\frac{M \cdot B}{I}} \]
Where:
- \( M \) is the magnetic moment,
- \( B \) is the magnetic field strength,
- \( I \) is the moment of inertia.

Since both magnets are of the same size and mass, we can assume that \( B \) and \( I \) are constant for both magnets.

### Step 4: Set up the ratio of frequencies
We can set up the ratio of the frequencies of the two magnets:
\[ \frac{f_1}{f_2} = \frac{M_1}{M_2} \]
Substituting the values of frequencies:
\[ \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{M_1}{M_2} \]

### Step 5: Simplify the ratio
Calculating the left-hand side:
\[ \frac{f_1}{f_2} = \frac{1}{6} \times \frac{4}{1} = \frac{4}{6} = \frac{2}{3} \]
Thus, we have:
\[ \frac{M_1}{M_2} = \frac{2}{3} \]

### Step 6: Find the ratio of magnetic moments
To find the ratio of the magnetic moments \( M_1 : M_2 \):
\[ M_1 : M_2 = 2 : 3 \]

### Conclusion
The ratio of the magnetic moments of the two magnets is \( 2:3 \).

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