Introduction:
Two oscillating simple pendulums with time periods T and 5T/4 are said to be "in phase" when they reach their maximum displacements in the same direction at the same time. In this scenario, we are given that the two pendulums are in phase at a given time.

Explanation:
To determine when the two pendulums will be in phase again, we need to find the time it takes for both pendulums to complete a full cycle and return to their initial positions.

Let's analyze the time period of each pendulum separately:

Time period of the first pendulum:
The time period T represents the time it takes for the first pendulum to complete one full cycle, from its maximum displacement in one direction, through its equilibrium position, to its maximum displacement in the opposite direction, and back to the equilibrium position.

Time period of the second pendulum:
The time period of the second pendulum is 5T/4, which means it takes this pendulum longer to complete a full cycle compared to the first pendulum. This implies that the second pendulum oscillates at a slower rate than the first pendulum.

Finding the time when both pendulums are in phase again:
To find the time when both pendulums are in phase again, we need to determine when both pendulums complete an integer number of cycles.

Since the time period of the second pendulum is longer, the number of cycles it completes in a given time will be less compared to the first pendulum.

Let's analyze the options provided:

1) 4T: This would mean both pendulums complete 4 cycles of the first pendulum's time period. However, the second pendulum completes only (4 * 5/4 = 5) cycles of its time period. Thus, the pendulums will not be in phase again after 4T.

2) 3T: This would mean both pendulums complete 3 cycles of the first pendulum's time period. However, the second pendulum completes only (3 * 5/4 = 15/4) cycles of its time period. Thus, the pendulums will not be in phase again after 3T.

3) 6T: This would mean both pendulums complete 6 cycles of the first pendulum's time period. However, the second pendulum completes only (6 * 5/4 = 15/2) cycles of its time period. Thus, the pendulums will not be in phase again after 6T.

4) 5T: This would mean both pendulums complete 5 cycles of the first pendulum's time period. The second pendulum completes (5 * 5/4 = 25/4) cycles of its time period. Since 25/4 is an integer (6.25), the pendulums will be in phase again after 5T.

Conclusion:
After analyzing the options, we can conclude that the two oscillating simple pendulums with time periods T and 5T/4 will be in phase again after an elapse of time 5T.

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