To determine the phase difference between two particles executing simple harmonic motion (SHM) with the same amplitude and frequency but separated by a distance along the x-axis, we can analyze their positions mathematically. Let's break this down step by step.
Understanding Simple Harmonic Motion
In SHM, the position of a particle can be described by the equation:
Here, A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant. For two particles, we can denote their positions as:
- x₁(t) = A sin(ωt) (for the first particle)
- x₂(t) = A sin(ωt + φ) (for the second particle)
Analyzing the Separation
The mean position of the two particles is separated by a distance X₀, which means that at any given time, the distance between them can be expressed as:
- Δx = x₂(t) - x₁(t) = A sin(ωt + φ) - A sin(ωt)
Using the sine subtraction formula, we can rewrite this as:
- Δx = A [sin(ωt + φ) - sin(ωt)]
Applying the identity for the difference of sines, we get:
- Δx = 2A cos( (ωt + φ + ωt)/2 ) sin(φ/2)
Maximum Separation Condition
The maximum separation between the two particles occurs when the sine function reaches its maximum value of 1. Therefore, we can express the maximum separation as:
According to the problem, the maximum separation is given as X₀ + A. Thus, we can set up the equation:
Rearranging gives us:
- 2A sin(φ/2) - A = X₀
- A(2 sin(φ/2) - 1) = X₀
Solving for Phase Difference
From this equation, we can isolate sin(φ/2):
- 2 sin(φ/2) = (X₀ + A)/A
- sin(φ/2) = (X₀ + A)/(2A)
Now, we know that X₀ > A, which implies that (X₀ + A)/(2A) > 1/2. The sine function achieves a value of 1/2 at an angle of 30 degrees. Therefore, we can conclude:
Final Result
Thus, the phase difference between the two particles is 60 degrees, which corresponds to option (2). This analysis shows how the separation and phase relationship in SHM can be mathematically derived, providing a clear understanding of the motion involved.
