 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/f/solution

ω = 3.95/0.10 = 6.28490 rad s^{- 1}, γ = \frac{0.05}{2 ( 0.10 )} = 0.2500 s^{- 1}, ν = 6.27993 rad s^{- 1} .

\frac{x ( t )}{c} \frac{y ( t )}{c} = e^{- γ t} [cos ν t + \frac{γ - ω}{ν} sin ν t], = e^{- γ t} [cos ν t + \frac{γ + ω}{ν} sin ν t] .

2 π / ν = 1.00052 s

gives almost four anticlockwise turns in four seconds. The radius envelope falls by

e^{- γ (4 s)} = e^{- 1} = 0.3679

. Substitution into the exact coordinates gives

(x (4 s), y (4 s)) = (0.37245 c, 0.36286 c), r (4 s) / r_{0} = 0.36769.

The left panel annotates the initial tangent, time marks and endpoint. Because

c

is an unspecified starting coordinate, the axes are normalized by

c

rather than assigned numerical lengths.

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