two particles undergoing simple harmonic motion have same amplitude and time period .if themaximum distance between them is equal to their amplitude,find phase difference between them

To tackle the problem of two particles undergoing simple harmonic motion (SHM) with the same amplitude and time period, we need to delve into the concept of phase difference. The phase difference between two oscillating particles can significantly affect their relative positions and motions.

Understanding the Basics of SHM

In simple harmonic motion, the position of a particle can be described by the equation:

• x(t) = A cos(ωt + φ)

Here, A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant. The phase constant determines the starting position of the particle in its oscillation cycle.

Setting Up the Problem

Let’s denote the two particles as Particle 1 and Particle 2. Since they have the same amplitude (A) and time period (T), we can express their positions as:

• x₁(t) = A cos(ωt + φ₁)
• x₂(t) = A cos(ωt + φ₂)

Here, φ₁ and φ₂ are the phase constants for Particle 1 and Particle 2, respectively.

Maximum Distance Between the Particles

The maximum distance between the two particles occurs when they are at their farthest points from each other. This distance can be expressed as:

• D = |x₁(t) - x₂(t)|

Substituting the expressions for x₁(t) and x₂(t), we get:

• D = |A cos(ωt + φ₁) - A cos(ωt + φ₂)|

Factoring out the amplitude, we have:

• D = A |cos(ωt + φ₁) - cos(ωt + φ₂)|

Using the Given Condition

According to the problem, the maximum distance between the two particles is equal to their amplitude (D = A). This leads us to:

• A |cos(ωt + φ₁) - cos(ωt + φ₂)| = A

Dividing both sides by A (assuming A ≠ 0), we find:

• |cos(ωt + φ₁) - cos(ωt + φ₂)| = 1

Analyzing the Cosine Function

The maximum value of the expression |cos(α) - cos(β)| is 1 when the two cosine values are at their extremes. This occurs when one cosine is 1 and the other is -1. Therefore, we can set up the following equations:

• cos(ωt + φ₁) = 1 and cos(ωt + φ₂) = -1

This implies:

• ωt + φ₁ = 2nπ (for some integer n)
• ωt + φ₂ = (2n + 1)π

Finding the Phase Difference

From the above equations, we can derive the phase difference:

• φ₂ - φ₁ = (2n + 1)π - 2nπ = π

This indicates that the phase difference between the two particles is:

• Δφ = φ₂ - φ₁ = π

Conclusion

The phase difference between the two particles undergoing simple harmonic motion, given that their maximum distance equals their amplitude, is π radians. This means that when one particle is at its maximum positive displacement, the other is at its maximum negative displacement, effectively being out of phase by half a cycle.