Two particles execute simple harmonic motion of the same amplitude and frequency along the same straight line. They pass one another when going in opposite directions each time their is half their amplitude. Find the phase difference b/w them.

To solve the problem of finding the phase difference between two particles executing simple harmonic motion (SHM), we need to analyze their motion in detail. Given that both particles have the same amplitude and frequency, we can express their positions mathematically and determine the conditions under which they pass each other at half their amplitude.

Understanding Simple Harmonic Motion

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

x(t) = A cos(ωt + φ)

Where:

• x(t) is the position of the particle at time t.
• A is the amplitude of the motion.
• ω is the angular frequency, related to the frequency f by the equation ω = 2πf.
• φ is the phase constant, which determines the starting position of the particle.

Setting Up the Equations

Let’s denote the two particles as Particle 1 and Particle 2. Their positions can be expressed as follows:

• For Particle 1: x₁(t) = A cos(ωt + φ₁)
• For Particle 2: x₂(t) = A cos(ωt + φ₂)

Condition for Passing Each Other

The problem states that the particles pass each other when they are at half their amplitude, which is A/2. This means we need to set up the equations:

• x₁(t) = A/2
• x₂(t) = -A/2

Finding the Phase Difference

Substituting the condition into the equations, we have:

• A cos(ωt + φ₁) = A/2
• A cos(ωt + φ₂) = -A/2

Dividing through by A gives:

• cos(ωt + φ₁) = 1/2
• cos(ωt + φ₂) = -1/2

The cosine function equals 1/2 at angles of π/3 and 5π/3, while it equals -1/2 at angles of 2π/3 and 4π/3.

Calculating the Phase Difference

Let’s assume:

• ωt + φ₁ = π/3 (for Particle 1)
• ωt + φ₂ = 2π/3 (for Particle 2)

Now, we can find the phase difference:

Δφ = φ₂ - φ₁

Substituting the values:

Δφ = (2π/3) - (π/3) = π/3

Thus, the phase difference between the two particles is π/3 radians.

Final Thoughts

This result indicates that the two particles are out of phase by 60 degrees, which is consistent with their respective positions at half amplitude when they pass each other. This analysis illustrates the beauty of simple harmonic motion and how phase relationships can be determined through mathematical reasoning.