A loaded spring vibrates with a period T. The spring is divided into four equal parts and the same load is divided into four equal parts and the same load is suspended from one end of these parts. The new period is

Options

A. T
B. 2 T
C. `T//2`
D. `T//4`

Explanation

To solve the problem step by step, we need to understand how the time period of a spring changes when it is divided into parts and how the load is applied.

### Step 1: Understand the original time period of the spring
The time period \( T \) of a spring-mass system is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass attached to the spring and \( k \) is the spring constant.

### Step 2: Analyze the situation after dividing the spring
When the spring is divided into four equal parts, each part will have a new spring constant. The spring constant \( k' \) of each piece is given by:
\[ k' = 4k \]
This is because the spring constant is inversely proportional to the length of the spring. When the length is reduced to one-fourth, the spring constant increases by a factor of four.

### Step 3: Determine the new mass
The load is also divided into four equal parts. Therefore, if the original mass was \( m \), each part will now have a mass of:
\[ m' = \frac{m}{4} \]

### Step 4: Calculate the new time period
Now, we can calculate the new time period \( T' \) using the new mass and the new spring constant:
\[ T' = 2\pi \sqrt{\frac{m'}{k'}} = 2\pi \sqrt{\frac{m/4}{4k}} = 2\pi \sqrt{\frac{m}{16k}} = 2\pi \frac{1}{4} \sqrt{\frac{m}{k}} = \frac{T}{4} \]
Thus, the new time period \( T' \) is one-fourth of the original time period \( T \).

### Conclusion
The new period \( T' \) is:
\[ T' = \frac{T}{4} \]

### Final Answer
The new period is \( \frac{T}{4} \).

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