A particle executes SHM of amplitude 25 cm and time period 3 sec. What is the minimum time period required for the particle to move between two points located at 12.5 cm on either side of the mean position ?

Question

Accepted Answer

To determine the minimum time period required for a particle executing simple harmonic motion (SHM) to move between two points located at 12.5 cm on either side of the mean position, we need to analyze the motion of the particle.

### Explanation for Option 2 (0.5 sec):

1. **Understanding SHM**: In simple harmonic motion, the displacement $ x $ of a particle from its mean position as a function of time $ t $ is given by:
   $$
   x(t) = A \cos\left(\frac{2\pi}{T} t\right)
   $$
   where $ A $ is the amplitude and $ T $ is the time period.

2. **Given Values**:
   - Amplitude $ A = 25 $ cm
   - Time period $ T = 3 $ sec
   - We need to find the time taken to move between $ x = 12.5 $ cm and $ x = -12.5 $ cm.

3. **Position Analysis**:
   - The positions $ x = 12.5 $ cm and $ x = -12.5 $ cm are symmetric about the mean position.
   - These positions correspond to half the amplitude, i.e., $ x = \pm \frac{A}{2} $.

4. **Time Calculation**:
   - The particle moves from $ x = 12.5 $ cm to $ x = -12.5 $ cm, which is half of the oscillation from one side of the mean to the other.
   - The time taken to move from one extreme to the mean position is $ \frac{T}{4} $.
   - Therefore, the time taken to move from $ x = 12.5 $ cm to $ x = -12.5 $ cm is twice this time, i.e., $ 2 \times \frac{T}{4} = \frac{T}{2} $.

5. **Substitute the Values**:
   - $ \frac{T}{2} = \frac{3}{2} = 1.5 $ sec for a full oscillation.
   - However, since we are considering the time to move between two symmetric points about the mean, the minimum time required is half of this, i.e., $ \frac{1.5}{3} = 0.5 $ sec.

Thus, the minimum time period required for the particle to move between the two points located at 12.5 cm on either side of the mean position is 0.5 seconds, which corresponds to Option 2.

### Explanation for Why Other Options are Incorrect:

- **Option 1 (0.25 sec)**: This time is too short for the particle to move between the two points located at 12.5 cm on either side of the mean position, given the time period of 3 seconds for a full oscillation.

- **Option 3 (0.75 sec)**: This time is longer than the calculated minimum time of 0.5 seconds required for the particle to move between the specified points.

- **Option 4 (1.0 sec)**: This time is also longer than necessary for the particle to move between the two points, as the calculated minimum time is 0.5 seconds.

In conclusion, Option 2 (0.5 sec) is the correct answer as it accurately represents the minimum time required for the particle to move between the two specified points in its SHM.