 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1/section-d/d12/b/solution

The transient is the homogeneous part of a damped harmonic oscillator,

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

α = b / (2 m)

and

ω_{0} = k / m

. Its characteristic roots are

- α \pm α^{2} - ω_{0}^{2}

For $α < ω_{0}$ , the underdamped response is $x = e^{- α t} (A cos ω_{d} t + B sin ω_{d} t)$ , with $ω_{d} = ω_{0}^{2} - α^{2}$ . The displacement oscillates with a decaying envelope.
At critical damping, $x = (A + Bt) e^{- ω_{0} t}$ . It has a repeated root, so two identical exponentials are not an independent pair of solutions.
For $α > ω_{0}$ , two real negative exponentials produce an overdamped return without sustained oscillation. In the strong-damping limit the slow decay rate is approximately $k / b$ , not $b / m$ .

Initial position and velocity determine A,B. For an imposed force or support motion, the full solution is a particular response plus this transient, which decays for b>0 and k>0. The mechanical energy obeys

d (K + U) / d t = - b \overset{x}{˙}^{2}

. In light damping its cycle-averaged envelope decays as

e^{- 2 α t}

, with

Q ≃ ω_{0} / (2 α)

. Initial conditions can still produce an overshoot in a nonoscillatory regime; the damping category describes the roots rather than every possible trajectory.

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