A particle is simultaneously acted upon bytwo collinear harmonic oscillations of samefrequency but different amplitudes anddifferent initial phases. Determine thedisplacement of the resultant oscillatio

To determine the displacement of the resultant oscillation, we need to consider the two collinear harmonic oscillations acting on the particle. Let's denote the amplitudes of the two oscillations as A1 and A2, and their initial phases as φ1 and φ2, respectively.

The equation for a harmonic oscillation is given by: x = A * cos(ωt + φ)

Where x is the displacement, A is the amplitude, ω is the angular frequency, t is the time, and φ is the initial phase.

Since the two oscillations have the same frequency, we can assume that their angular frequencies are equal, denoted as ω1 = ω2 = ω.

Now, let's consider the resultant oscillation. The displacement of the resultant oscillation at any given time t can be found by adding the displacements of the two individual oscillations at that time.

x_resultant = x1 + x2

Using the equation for harmonic oscillation, we can write:

x1 = A1 * cos(ωt + φ1) x2 = A2 * cos(ωt + φ2)

Substituting these equations into the equation for the resultant oscillation, we get:

x_resultant = A1 * cos(ωt + φ1) + A2 * cos(ωt + φ2)

To simplify this equation, we can use the trigonometric identity:

cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)

Applying this identity to the equation for the resultant oscillation, we get:

x_resultant = A1 * cos(ωt) * cos(φ1) - A1 * sin(ωt) * sin(φ1) + A2 * cos(ωt) * cos(φ2) - A2 * sin(ωt) * sin(φ2)

Now, we can rearrange the terms and combine the coefficients of cos(ωt) and sin(ωt):

x_resultant = (A1 * cos(φ1) + A2 * cos(φ2)) * cos(ωt) - (A1 * sin(φ1) + A2 * sin(φ2)) * sin(ωt)

Let's denote the coefficients of cos(ωt) and sin(ωt) as B1 and B2, respectively:

B1 = A1 * cos(φ1) + A2 * cos(φ2) B2 = A1 * sin(φ1) + A2 * sin(φ2)

Now, we can rewrite the equation for the resultant oscillation as:

x_resultant = B1 * cos(ωt) - B2 * sin(ωt)

This equation represents a harmonic oscillation with an amplitude of √(B1^2 + B2^2) and an initial phase of arctan(B2/B1).

Therefore, the displacement of the resultant oscillation is given by:

x_resultant = √(B1^2 + B2^2) * cos(ωt + φ_resultant)

where φ_resultant = arctan(B2/B1).

By calculating B1 and B2 using the given values of A1, A2, φ1, and φ2, you can determine the displacement of the resultant oscillation at any given time t.