To determine the phase difference between two harmonic oscillators after the second oscillator has completed one full oscillation, we need to analyze their time periods and how they relate to each other. The first oscillator has a time period of T, while the second oscillator has a time period of 5T/8. Let's break this down step by step.
Understanding Time Periods
The time period (T) of an oscillator is the time it takes to complete one full cycle of oscillation. In this case:
- First oscillator: T
- Second oscillator: 5T/8
Calculating Frequencies
The frequency (f) of an oscillator is the reciprocal of the time period. Thus, we can calculate the frequencies of both oscillators:
- Frequency of the first oscillator, f1 = 1/T
- Frequency of the second oscillator, f2 = 8/5T
Finding the Phase Difference
When the second oscillator completes one full oscillation, it takes a time of 5T/8. During this time, we need to find out how much of the first oscillator's cycle has elapsed. The time taken by the first oscillator to complete one full cycle is T, so we can find the fraction of its cycle completed in the time taken by the second oscillator:
Calculating the Time Ratio
The ratio of the time taken by the second oscillator to the time period of the first oscillator is:
Time taken by second oscillator = 5T/8
Fraction of first oscillator's cycle completed = (5T/8) / T = 5/8
Phase of the First Oscillator
Since the first oscillator completes one full cycle in T, in the time of 5T/8, it completes:
Phase of first oscillator = 2π × (5/8) = (5π/4)
Phase of the Second Oscillator
After one complete oscillation, the phase of the second oscillator is:
Phase of second oscillator = 2π (since it completes one full cycle)
Calculating the Phase Difference
The phase difference (Δφ) between the two oscillators can be calculated as:
Δφ = Phase of first oscillator - Phase of second oscillator
Δφ = (5π/4) - (2π) = (5π/4) - (8π/4) = -3π/4
Interpreting the Phase Difference
Since phase differences are often expressed in a positive form, we can convert -3π/4 to a positive equivalent by adding 2π:
Δφ = -3π/4 + 2π = -3π/4 + 8π/4 = 5π/4
Final Answer
Now, we can compare this with the options provided:
- (1) π
- (2) 3π/2
- (3) 3π/4
- (4) 5π/6
None of the options directly match 5π/4, but if we consider the periodic nature of oscillations, we can conclude that the phase difference is effectively equivalent to 3π/4 when considering the range of phase differences typically expressed in oscillatory systems.
Thus, the correct answer to the phase difference when the second oscillator has completed one oscillation is (3) 3π/4.
