A magnet has time period of oscillations T in earth's magnetic field at a place. If it is replaced by a magnet of magnetic moment six times the magnetic moment of the original one, the time period at the same place will be

Options

A. `T //6`
B. `6T`
C. `sqrt(6T)`
D. `T // sqrt(6)`

Answer

To solve the problem, we need to analyze how the time period of oscillation of a magnet in a magnetic field changes when the magnetic moment of the magnet is altered.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
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 of the magnet,
- \( m \) is the magnetic moment of the magnet,
- \( B \) is the magnetic field strength.

2. **Identifying the Initial Conditions**:
Let the initial magnetic moment be \( m_1 = m \) and the time period be \( T_1 = T \). Thus, we can express the time period as:
\[ T_1 = 2\pi \sqrt{\frac{I}{mB}} \]

3. **Changing the Magnetic Moment**:
When the magnet is replaced with a new magnet that has a magnetic moment \( m_2 = 6m \), we need to find the new time period \( T_2 \).

4. **Expressing the New Time Period**:
The new time period can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{I}{m_2 B}} = 2\pi \sqrt{\frac{I}{6mB}} \]

5. **Relating the New Time Period to the Old One**:
We can relate \( T_2 \) to \( T_1 \):
\[ T_2 = 2\pi \sqrt{\frac{I}{6mB}} = 2\pi \sqrt{\frac{1}{6}} \sqrt{\frac{I}{mB}} = \frac{T_1}{\sqrt{6}} \]
Thus, substituting \( T_1 = T \):
\[ T_2 = \frac{T}{\sqrt{6}} \]

6. **Final Result**:
Therefore, the time period of the new magnet in the same magnetic field is:
\[ T_2 = \frac{T}{\sqrt{6}} \]

### Conclusion:
The time period at the same place will be \( \frac{T}{\sqrt{6}} \).