One mass m is suspended from a spring. Time period of oscilation is T. now if spring is divided into n pieces & these are joined in parallel order then time period of oscillation if same mass is suspended.

Options

A. `2pisqrt((n^(2)m)/k)`
B. `2pisqrt((n^(2)k)/m)`
C. `2pisqrt(m/(n^(2)k))`
D. `2pisqrt(k/(n^(2)m))`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Time Period
The time period \( T \) of a mass \( m \) suspended from a spring with spring constant \( k \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

### Step 2: Analyze the Spring After Cutting
When the spring is cut into \( n \) equal pieces, the length of each piece becomes \( \frac{L}{n} \). The spring constant \( k' \) of each piece can be derived from the relationship between the spring constant and the length of the spring. The product of the length and spring constant remains constant:
\[ L \cdot k = \left(\frac{L}{n}\right) \cdot k' \]
From this, we can solve for \( k' \):
\[ k' = n \cdot k \]

### Step 3: Determine the Effective Spring Constant
When these \( n \) springs are joined in parallel, the effective spring constant \( k_{\text{eff}} \) is the sum of the spring constants of each piece:
\[ k_{\text{eff}} = k' + k' + \ldots + k' \quad (n \text{ times}) \]
This simplifies to:
\[ k_{\text{eff}} = n \cdot k' = n \cdot (n \cdot k) = n^2 \cdot k \]

### Step 4: Calculate the New Time Period
Now, we can find the new time period \( T' \) when the mass \( m \) is suspended from the new effective spring constant \( k_{\text{eff}} \):
\[ T' = 2\pi \sqrt{\frac{m}{k_{\text{eff}}}} = 2\pi \sqrt{\frac{m}{n^2 k}} \]
This can be simplified to:
\[ T' = \frac{2\pi}{n} \sqrt{\frac{m}{k}} \]

### Step 5: Relate the New Time Period to the Original Time Period
Since the original time period \( T \) is \( 2\pi \sqrt{\frac{m}{k}} \), we can express \( T' \) in terms of \( T \):
\[ T' = \frac{T}{n} \]

### Final Answer
Thus, the new time period of oscillation when the spring is divided into \( n \) pieces and joined in parallel is:
\[ T' = \frac{T}{n} \]