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

### Step 1: Understand the given data
We have two magnets of the same size and mass. The first magnet makes 10 oscillations per minute, and the second magnet makes 15 oscillations per minute.

### Step 2: Convert oscillations per minute to time period
The time period \( T \) (in seconds) is the reciprocal of the frequency (oscillations per minute).

For the first magnet:
\[ T_1 = \frac{60 \text{ seconds}}{10 \text{ oscillations}} = 6 \text{ seconds} \]

For the second magnet:
\[ T_2 = \frac{60 \text{ seconds}}{15 \text{ oscillations}} = 4 \text{ seconds} \]

### Step 3: Relate the time periods to magnetic moments
The time period \( T \) of a magnet oscillating in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{MB}} \]
Where:
- \( I \) is the moment of inertia (which is constant for both magnets since they are of the same size and mass),
- \( M \) is the magnetic moment,
- \( B \) is the magnetic field (assumed constant).

### Step 4: Set up the ratio of time periods
Since the moment of inertia \( I \) and the magnetic field \( B \) are constants, we can set up the following ratio:
\[ \frac{T_1}{T_2} = \sqrt{\frac{M_1}{M_2}} \]

### Step 5: Square both sides
Squaring both sides gives:
\[ \left(\frac{T_1}{T_2}\right)^2 = \frac{M_1}{M_2} \]

### Step 6: Substitute the time periods
Substituting the values of \( T_1 \) and \( T_2 \):
\[ \frac{T_1}{T_2} = \frac{6}{4} = \frac{3}{2} \]
Squaring this gives:
\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 7: Find the ratio of magnetic moments
Thus, we have:
\[ \frac{M_1}{M_2} = \frac{9}{4} \]
This implies:
\[ \frac{M_2}{M_1} = \frac{4}{9} \]

### Conclusion
The ratio of the magnetic moments \( M_2 : M_1 \) is \( 4 : 9 \).