Past exam of the physics A course of the University of Cambridge / 2017 / Paper 2 / Section B / B6 / Solution

Use the $e^{i\omega t}$ convention for the harmonic force and displacement complex amplitudes. Substitution in the forced harmonic oscillator equation givesIts magnitude and phase relative to the force areAt zero frequency, $|R|=1/k$ and $\varphi =0$. At ${\omega }_0=\sqrt{k/m}$, $\varphi =-\pi /2$; at high frequency the phase approaches $-\pi$. For light viscous damping, the quality factor $Q=m{\omega }_0/b$ is large and there is a tall narrow displacement peak close to ${\omega }_0$.

To find the displacement peak, minimize the squared denominator:The nonzero minimum givesFor stronger damping there is no positive-frequency displacement maximum. When $\omega \gg {\omega }_0$ and $b\ll m\omega$,The inertia then dominates the restoring and damping forces.

The four supporting springs act in parallel, so their total spring constant is $k=4.00\times 10^4\mathrm{N}{\mathrm{m}}^{-1}$. The driving angular frequency is $\omega =2\pi (10)=62.83{\mathrm{s}}^{-1}$. The force-driven vibration isolation estimate givesAt this mass, $f_0=\sqrt{k/m}/(2\pi )=0.894\mathrm{Hz}\ll 10\mathrm{Hz}$, and $Q=\sqrt{mk}/b\simeq 14.2$, so the assumptions are consistent. The exact forced amplitude isIt is indeed close to $1\mu \mathrm{m}$, but it is slightly above the strict requested bound. Setting the exact expression to $10^{-6}\mathrm{m}$ gives the high-mass thresholdChoose a mass strictly greater than this threshold for a strict inequality; $1.28\times 10^3\mathrm{kg}$ is a suitable rounded choice. Simply rounding the high-frequency estimate down would not guarantee the bound.

Measure downward displacement relative to the former mean equilibrium. The added $M=20\mathrm{kg}$ changes the equilibrium byAssume the load is deposited without a significant impact impulse and then moves with the table. The moving mass becomes $m+M$. Ignoring the tiny pre-existing $10\mathrm{Hz}$ vibration in the sketch, the added-load step response of a supported table isFor the estimated mass, $\beta =0.194{\mathrm{s}}^{-1}$, the envelope time is $1/\beta =5.15\mathrm{s}$, and the period is about $1.13\mathrm{s}$. The table overshoots and rings about the new equilibrium, with the displacement starting continuously at zero and with zero initial slope. A real impact adds an initial-velocity transient; the problem gives no impact history. The residual driven vibration, of order a micrometre, can be superposed.

To settle quickly after loading, critical damping is preferable to the supplied weak damper:Using the strict-bound mass instead changes this by less than $1\%$. This eliminates free ringing while avoiding the slow tail of severe overdamping. In the stipulated external-force model, greater damping also reduces the steady forced amplitude; there is no finite optimum based solely on that amplitude. A critical or near-critical value is a practical compromise for rapid settling. If the disturbance were specified as floor displacement transmitted through the legs, damping would introduce an additional transmission tradeoff, but that is a different forcing model.