A thin magnetic needle oscillates in a horizontal plane with a period T. It is broken into n equals parts. The time period of each part will be

Options

A. T/n
B. T
C. T n
D. `1//Tn`

Answer

To solve the problem of determining the time period of each part of a broken magnetic needle, we can follow these steps:

### Step 1: Understand the Initial Conditions
The original magnetic needle has a period \( T \). The magnetic moment \( M \) of the needle is defined, and its length is \( L \). The mass of the needle is \( m \).

### Step 2: Write the Formula for Time Period
The time period \( T \) of the magnetic needle is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{m \cdot B}} \]
where \( I \) is the moment of inertia, \( m \) is the mass, and \( B \) is the magnetic field.

### Step 3: Calculate the Moment of Inertia
For a thin rod (the magnetic needle) rotating about its center, the moment of inertia \( I \) is given by:
\[ I = \frac{mL^2}{12} \]

### Step 4: Substitute the Moment of Inertia into the Time Period Formula
Substituting \( I \) into the time period formula gives:
\[ T = 2\pi \sqrt{\frac{\frac{mL^2}{12}}{m \cdot B}} = 2\pi \sqrt{\frac{L^2}{12B}} \]

### Step 5: Break the Needle into \( n \) Equal Parts
When the needle is broken into \( n \) equal parts:
- Each part will have a length of \( \frac{L}{n} \).
- The mass of each part will be \( \frac{m}{n} \).

### Step 6: Calculate the New Moment of Inertia for Each Part
The moment of inertia for each part, which is still a thin rod, is:
\[ I' = \frac{\frac{m}{n} \left(\frac{L}{n}\right)^2}{12} = \frac{mL^2}{12n^3} \]

### Step 7: Substitute the New Values into the Time Period Formula
The new time period \( T' \) for each part will be:
\[ T' = 2\pi \sqrt{\frac{I'}{\frac{m}{n} \cdot B}} = 2\pi \sqrt{\frac{\frac{mL^2}{12n^3}}{\frac{m}{n} \cdot B}} = 2\pi \sqrt{\frac{L^2}{12n^2B}} \]

### Step 8: Relate the New Time Period to the Original Time Period
From the original time period \( T \):
\[ T' = \frac{T}{n} \]

### Conclusion
Thus, the time period of each part after breaking the needle into \( n \) equal parts is:
\[ T' = \frac{T}{n} \]