The time period of a mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be:

Options

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

Explanation

To solve the problem, we need to understand how the time period of a mass-spring system changes when the spring is cut into smaller parts. Here’s a step-by-step solution:

### Step 1: Understand the time period of a spring-mass system
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: Determine the spring constant of the original spring
Let the spring constant of the original spring be \( k \). The time period of the system is given as \( T \).

### Step 3: Cut the spring into four equal parts
When the spring is cut into four equal parts, each part will have a new spring constant. The spring constant of a spring is inversely proportional to its length. If the original length of the spring is \( l \), each part will have a length of \( \frac{l}{4} \).

### Step 4: Calculate the new spring constant
The relationship between the spring constant \( k \) and the length \( l \) of the spring is given by:
\[ k \cdot l = k' \cdot \frac{l}{4} \]
where \( k' \) is the new spring constant for one of the parts. Rearranging gives:
\[ k' = 4k \]

### Step 5: Calculate the new time period
Now, we need to find the time period \( T' \) for the mass \( m \) suspended from one of the new springs with spring constant \( k' = 4k \):
\[ T' = 2\pi \sqrt{\frac{m}{k'}} = 2\pi \sqrt{\frac{m}{4k}} = \frac{1}{2} \cdot 2\pi \sqrt{\frac{m}{k}} = \frac{T}{2} \]

### Conclusion
Thus, the new time period \( T' \) when the mass is suspended from one of the four equal parts of the spring is:
\[ T' = \frac{T}{2} \]

### Final Answer
The new time period will be \( \frac{T}{2} \).

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