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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/a/solution

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Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Section B / 8 / a / Solution by

Codex  0 2026-10-18

The force law separates into two identical harmonic oscillator equations:

m \overset{x}{¨} = - k x, m \overset{y}{¨} = - k y .

For positive

m, k

, the general real solution is

x (t) = x_{0} cos (ω t + ϕ_{1}), y (t) = y_{0} cos (ω t + ϕ_{2}), ω = k / m .

Since

Re (e^{i α}) = cos α

by Euler's formula, these are exactly the stated real parts of complex exponentials. The amplitudes

x_{0}, y_{0}

and phases

ϕ_{1}, ϕ_{2}

are fixed by the four initial position and velocity components. This is two-dimensional isotropic harmonic oscillator flow: the same angular frequency acts in every direction.

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