Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section C

After switching, Kirchhoff's voltage law gives the free series RLC circuit equationIts characteristic roots are $-\gamma \pm i{\omega }_d$, whereA real, nonzero oscillation frequency requires $R^2<4L/C$. For a passive resistor,The endpoint $R=0$ is the undamped limit; strictly decaying oscillations have $0<R<894\Omega$. The critical endpoint itself does not have a nonzero ${\omega }_d$.

Let $Q=C{V}_0$ denote the initial electric charge, so $Q=30\mu \mathrm{C}$ and $\dot{q}(0)=0$. Distinguish $Q$ from the amplitude coefficient $q_0$ in an exponential-times-cosine representation: imposing zero initial electric current generally requires a nonzero phase.

Put ${\omega }_0^2=1/(LC)$. The initial conditions giveThe printed electric current arrow points toward the positively labelled capacitor plate, so its signed electric current is $I=\dot{q}$, not the positive discharge magnitude. HenceFor the stated electrical resistance,The voltage starts at $6\mathrm{V}$ with zero slope; the signed electric current starts at zero and initially becomes negative. Its initial slope is $\dot{I}(0)=-V_0/L=-6\mathrm{A}{\mathrm{s}}^{-1}$. Both oscillate with an $e^{-25t}$ envelope and a $40\mathrm{ms}$ amplitude time constant. The phase difference is close to a quarter period, with the small damping correction displayed explicitly in $V_C$.

The capacitor stores electrostatic energy, the inductor stores magnetic energy, and the resistor converts energy into heat. Their instantaneous values areThe sign convention for electric current does not change these expressions. With the initial voltage, $E_C(0)=C{V}_0^2/2=90\mu \mathrm{J}$ and $E_I(0)=P_R(0)=0$.

The two stored energies follow directly by squaring the expressions for electric charge and electric current:Both are nonnegative, with alternating maxima. Initially all stored energy is in the capacitor; at a zero of $q$ it is entirely in the inductor. The total isThe total energy decreases monotonically, with a horizontal tangent whenever the electric current is zero. Its envelope decays on the $1/(2\gamma )=20\mathrm{ms}$ scale, but the exact total is not a featureless exponential: the resistor's instantaneous loss varies within each oscillation. The figure in part (a) shows this gradual modulation of the downward slope as well as the component energies.

Measure the mass's displacement $x$ from its static equilibrium height. This removes the constant spring extension supporting its weight. If the spring's top moves with the box, its incremental extension is $y_2-x$, so its incremental upward force is $k(y_2-x)$. The specified damping force is against the mass's velocity in the static frame. Newton's second law therefore givesIn absolute height coordinates an additional constant on the right expresses the spring's rest length, its fixed attachment height and gravity; translating to static equilibrium removes it. Replacing the given damping by $-b(\dot{x}-{\dot{y}}_2)$ would describe a different base-excited oscillator.

After the decaying transient, write the steady-state response as $x=A(\omega )\cos (\omega t-\Phi )$. Using a complex sinusoid givessoFor positive frequency, the phase lag lies between zero and $\pi$, passing through $\pi /2$ at ${\omega }_0=\sqrt{k/m}$. For small $b$, the frequency response begins at $a_s$, develops a narrow resonance near ${\omega }_0$, and falls asThe precise amplitude-peak frequency is $\sqrt{k/m-b^2/(2m^2)}$ when that expression is positive.

The pen records the relative displacement $z=x-y_2$. Its complex transfer function isAt high frequency, $x$ is small while $z\simeq -y_2$. The inertial mass then stays nearly stationary and the recorded relative motion reproduces the ground displacement, with reversed sign. At low frequency the mass follows the box and their relative motion is small; near resonance the response is strongly frequency-dependent. Here “high” means compared with the suspension's natural frequency, rather than an absolute seismic-frequency threshold.