 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-2/section-b/b6/a/solution

Substituting

x = A e^{pt}

p^{2} + γ p + ω_{0}^{2} = 0, p_{\pm} = - \frac{γ}{2} \pm \frac{γ ^{2}}{4} - ω_{0}^{2} .

For nonnegative viscous damping, the cases are

γ > 2 ω_{0} γ = 2 ω_{0} 0 \leq γ < 2 ω_{0} : x = C_{+} e^{p_{+} t} + C_{-} e^{p_{-} t} : x = (C_{1} + C_{2} t) e^{- ω_{0} t} : x = e^{- γ t /2} [C_{1} cos ω_{d} t + C_{2} sin ω_{d} t] overdamping, critical damping, underdamping, ω_{d} = ω_{0}^{2} - γ^{2} /4 .

The PDF's request to express these conditions in terms of

γ / m

is dimensionally inconsistent with its definition

γ = b / m

γ

already has units

s^{- 1}

. With the unnormalised equation

m \overset{x}{¨} + b \overset{x}{˙} + k x = 0

, the equivalent comparisons are

b / m ≷ 2 ω_{0}

, or

b^{2} ≷ 4 mk

. Also

ω_{0}

is the undamped natural angular frequency, not in general the driven displacement peak.

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