Two particles undergoing SHM are matched in frequency and amplitude but they differ in phase by 90 degree. When shallthey have the same displacement?

To understand when two particles undergoing simple harmonic motion (SHM) with the same frequency and amplitude but differing in phase by 90 degrees will have the same displacement, we need to delve into the mathematical representation of SHM and the implications of phase differences.

Understanding Simple Harmonic Motion

Simple harmonic motion can be described by the equation:

x(t) = A cos(ωt + φ)

In this equation:

• x(t) is the displacement at time t.
• A is the amplitude of the motion.
• ω is the angular frequency.
• φ is the phase constant.

Phase Difference Explained

When two particles are said to differ in phase by 90 degrees (or π/2 radians), their phase constants can be represented as:

• Particle 1: φ₁ = 0
• Particle 2: φ₂ = π/2

Thus, the displacement equations for the two particles become:

• Particle 1: x₁(t) = A cos(ωt)
• Particle 2: x₂(t) = A cos(ωt + π/2) = A sin(ωt)

Finding Common Displacement

To determine when both particles have the same displacement, we set their displacement equations equal to each other:

A cos(ωt) = A sin(ωt)

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

cos(ωt) = sin(ωt)

Solving the Equation

This equation can be solved by recognizing that the cosine and sine functions are equal at specific angles. The general solution for this equality occurs when:

ωt = π/4 + nπ (where n is any integer)

This means that the two particles will have the same displacement at:

• t = (π/4 + nπ)/ω

Visualizing the Motion

To visualize this, think of the unit circle. The cosine function represents the x-coordinate, while the sine function represents the y-coordinate. At angles of 45 degrees (π/4 radians) and 225 degrees (5π/4 radians), both coordinates are equal, which corresponds to the points in time when the displacements of the two particles match.

Conclusion

In summary, two particles undergoing SHM with the same frequency and amplitude but differing in phase by 90 degrees will have the same displacement at specific intervals defined by the equation t = (π/4 + nπ)/ω. This relationship highlights the fascinating interplay between phase, frequency, and displacement in harmonic motion.