 Solution

ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/section-b/8/b/solution

Put

δ = ϕ_{2} - ϕ_{1}

. If both coordinate amplitudes are nonzero, eliminate the common phase to obtain

(\frac{x}{x _{0}})^{2} + (\frac{y}{y _{0}})^{2} - 2 \frac{x y}{x _{0} y _{0}} cos δ = sin^{2} δ .

For

sin δ \neq = 0

, this is an ellipse centred at the origin. The following special cases exhaust the possible nontrivial shapes:

$δ = 0$ or $π$ gives straight-line simple harmonic motion, including an axis-aligned line if one coordinate amplitude vanishes.
$δ = \pm π /2$ gives an axis-aligned ellipse; if additionally $x_{0} = y_{0}$ , it is a circle traversed in one of the two orientations.
Other phase differences give tilted ellipses.

The degenerate zero-amplitude solution is the stationary origin. There are no general multi-loop patterns here because the two angular frequencies are equal. Another useful description is

r = a cos ω t + b sin ω t

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