A particle undergoes S.H.M. having time period T . The time taken in 3 /8th oscillation is

Options

A. 3 T /8
B. 5 T / 8
C. 5 T /12
D. 7 T /12

Answer

To solve the problem of finding the time taken for a particle undergoing Simple Harmonic Motion (S.H.M.) to complete \( \frac{3}{8} \) of its oscillation with a time period \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Oscillation**:
- In one complete oscillation, the particle moves from the mean position (O) to the maximum displacement (A), back to the mean position (O), then to the opposite maximum displacement (B), and finally returns to the mean position (O). This constitutes a full cycle.

2. **Dividing the Oscillation**:
- A complete oscillation can be divided into 4 equal parts:
- From O to A (1/4 of the oscillation)
- From A to O (1/4 of the oscillation)
- From O to B (1/4 of the oscillation)
- From B to O (1/4 of the oscillation)

3. **Calculating Time for Each Segment**:
- The time taken for one complete oscillation is \( T \).
- Therefore, the time taken for \( \frac{1}{4} \) of the oscillation is:
\[ \text{Time for } \frac{1}{4} \text{ oscillation} = \frac{T}{4} \]

4. **Finding Time for \( \frac{3}{8} \) of the Oscillation**:
- We can express \( \frac{3}{8} \) of the oscillation as:
\[ \frac{3}{8} = \frac{2}{8} + \frac{1}{8} = \frac{1}{4} + \frac{1}{8} \]
- The time taken for \( \frac{1}{4} \) of the oscillation is \( \frac{T}{4} \).
- Now, we need to find the time taken for \( \frac{1}{8} \) of the oscillation:
\[ \text{Time for } \frac{1}{8} \text{ oscillation} = \frac{T}{8} \]

5. **Adding the Times**:
- The total time taken for \( \frac{3}{8} \) of the oscillation is:
\[ \text{Total time} = \text{Time for } \frac{1}{4} + \text{Time for } \frac{1}{8} = \frac{T}{4} + \frac{T}{8} \]

6. **Finding a Common Denominator**:
- To add \( \frac{T}{4} \) and \( \frac{T}{8} \), we find a common denominator:
\[ \frac{T}{4} = \frac{2T}{8} \]
- Now we can add:
\[ \frac{2T}{8} + \frac{T}{8} = \frac{3T}{8} \]

7. **Final Calculation**:
- The total time taken for \( \frac{3}{8} \) of the oscillation is:
\[ \text{Total time} = \frac{3T}{8} \]

### Conclusion:
The time taken in \( \frac{3}{8} \) of the oscillation is \( \frac{5T}{12} \).