Given:

Two pendulums start swinging in phase.

They swing in phase after an interval of 18 seconds from the start.

To find:

Period of the simple pendulum.

Solution:

Let the period of the simple pendulum be T seconds.

Let the period of the second pendulum be P seconds.

Step 1: Find the ratio of periods of the pendulums.

When the pendulums swing in phase, the ratio of their periods is equal to the ratio of the time intervals after which they swing in phase.

Hence,

T/P = 18/P (After 18 seconds, both pendulums swing in phase)

Step 2: Solve for T.

Cross-multiplying the equation, we get

T = 18

Answer:

The period of the simple pendulum is 18 seconds.

Explanation:

When two pendulums start swinging in phase, it means that they start swinging together and their oscillations are synchronized. This happens when both pendulums are at their mean positions at the same time.

When the pendulums swing in phase again after 18 seconds from the start, it means that both pendulums have completed an integral number of oscillations in that time interval. Let n be the number of oscillations completed by the simple pendulum in this time interval.

Then, the number of oscillations completed by the second pendulum in this time interval is also n, as both pendulums swing in phase.

Therefore, we have

T * n = 18
P * n = 18


Dividing the second equation by the first equation, we get

T/P = 1


Since the two pendulums have the same phase at the end of the time interval, their phases must also be the same at the beginning of the time interval. Therefore, the simple pendulum must have completed an integral number of oscillations in the time interval. This number of oscillations is given by n = 18/T.

Substituting this value of n in the first equation, we get

T * (18/T) = 18


which simplifies to T = 18.

Hence, the period of the simple pendulum is 18 seconds.