A mass m is suspended from a spring of force constant k. The period of oscillation is `T_0` The spring is cut into 4 equal parts: The same mass m is suspended from one of the parts. What is the new period?

Explanation

To solve the problem, we need to find the new period of oscillation when a mass \( m \) is suspended from one of the four equal parts of a spring that originally had a spring constant \( k \).

### Step-by-Step Solution:

1. **Understand the Original Period**:
The period of oscillation \( T_0 \) for a mass \( m \) suspended from a spring with spring constant \( k \) is given by the formula:
\[ T_0 = 2\pi \sqrt{\frac{m}{k}} \]

2. **Determine the Spring Constant of the Cut Spring**:
When the spring is cut into 4 equal parts, the spring constant \( k' \) of each part increases. The relationship for the spring constant when a spring is cut into \( n \) equal parts is:
\[ k' = n \cdot k \]
Here, \( n = 4 \), so:
\[ k' = 4k \]

3. **Calculate the New Period**:
The new period \( T' \) when the mass \( m \) is suspended from one of the parts (with spring constant \( k' = 4k \)) is given by:
\[ T' = 2\pi \sqrt{\frac{m}{k'}} \]
Substituting \( k' \):
\[ T' = 2\pi \sqrt{\frac{m}{4k}} = 2\pi \sqrt{\frac{m}{k}} \cdot \frac{1}{2} \]
This simplifies to:
\[ T' = \frac{1}{2} T_0 \]

4. **Conclusion**:
The new period \( T' \) is half of the original period \( T_0 \):
\[ T' = \frac{T_0}{2} \]

### Final Answer:
The new period of oscillation when the mass \( m \) is suspended from one of the parts of the spring is \( \frac{T_0}{2} \).