If the particle repeats its motion after a fixed time interval of 8 s then after how much time its maximum value of PE will be attained after attaining its minimum value ?

Options

A. 2s
B. 4s
C. 8s
D. 1s

Answer

To solve the problem step by step, we need to analyze the motion of a particle undergoing simple harmonic motion (SHM) and determine the time taken to reach maximum potential energy after reaching minimum potential energy.

### Step 1: Understand the Time Period
The problem states that the particle repeats its motion after a fixed time interval of 8 seconds. This means that the time period (T) of the particle's motion is 8 seconds.

**Hint:** The time period is the total time taken for one complete cycle of motion.

### Step 2: Identify Minimum and Maximum Potential Energy Positions
In SHM, the potential energy (PE) is minimum at the mean position (x = 0) and maximum at the extreme positions (x = ±A, where A is the amplitude).

**Hint:** Remember that potential energy is minimum at the mean position and maximum at the extremes.

### Step 3: Determine the Time to Move from Mean to Extreme Position
To find the time taken to go from the mean position to the extreme position, we can relate it to the motion of a particle in uniform circular motion.

In SHM, the particle moves from the mean position to the extreme position (90 degrees in circular motion). This corresponds to a quarter of the full oscillation.

**Hint:** The time taken to go from the mean position to the extreme position is one-fourth of the time period.

### Step 4: Calculate the Time Taken
Since the time period T is 8 seconds, the time taken to move from the mean position to the extreme position is:

\[ \text{Time taken} = \frac{T}{4} = \frac{8 \text{ s}}{4} = 2 \text{ s} \]

**Hint:** Use the formula \( \text{Time} = \frac{T}{4} \) to find the time taken for a quarter of the oscillation.

### Conclusion
After reaching the minimum potential energy at the mean position, the maximum potential energy will be attained after 2 seconds.

**Final Answer:** The maximum value of potential energy will be attained after 2 seconds.