To determine when two pendulums with different natural frequencies will come back into phase after being released simultaneously, we need to consider their oscillatory behavior. When two oscillating systems are released in phase, they will only align again after a certain time interval, which depends on their frequencies. Let's break this down step by step.
Understanding Frequency and Phase
The natural frequency of a pendulum, denoted as f, is the rate at which it oscillates. If we have two pendulums with frequencies f1 and f2, they will complete their cycles in different amounts of time. The time period T for each pendulum is the reciprocal of its frequency:
- For the first pendulum: T1 = 1/f1
- For the second pendulum: T2 = 1/f2
Finding the Time Interval for Re-Alignment
To find when both pendulums will be in phase again, we need to determine the least common multiple (LCM) of their time periods. This is because the LCM will give us the smallest time interval at which both pendulums complete an integer number of cycles and thus align again.
The time interval at which they will come back into phase can be expressed as:
Time Interval = LCM(T1, T2)
Calculating the LCM
Now, substituting the time periods into the LCM formula, we have:
Time Interval = LCM(1/f1, 1/f2)
This means that the correct answer to your question is the least common multiple of the two time periods, which corresponds to option 4: lcm of 1/f1 and 1/f2.
Example for Clarity
Let’s illustrate this with an example. Suppose:
- f1 = 2 Hz (which gives T1 = 1/2 = 0.5 seconds)
- f2 = 3 Hz (which gives T2 = 1/3 ≈ 0.33 seconds)
To find the LCM of these two time periods, we can convert them to a common denominator:
- T1 = 0.5 = 3/6 seconds
- T2 = 0.33 ≈ 2/6 seconds
The LCM of 3/6 and 2/6 is 1 second, meaning both pendulums will be back in phase after 1 second.
Final Thoughts
In summary, when dealing with oscillating systems like pendulums, the time interval for them to come back into phase is determined by the least common multiple of their respective time periods. This approach ensures that both pendulums complete whole cycles and align perfectly again.
