Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Section A

Take a representative tennis-ball mass of $m\simeq 0.06\mathrm{kg}$. The speed is nonrelativistic, so the de Broglie wavelength isThis is an estimate because the mass was not specified. The wavelength is vastly smaller than any practicable aperture or obstacle separation: a diffraction angle of order $\lambda /d$ is unresolvably small for laboratory-scale $d$. Moreover, interactions with air, thermal radiation and the apparatus destroy observable coherence between macroscopically distinct paths. Quantum mechanics still applies; the small wavelength and rapid decoherence explain why ordinary experiments reveal classical trajectories rather than a measurable interference pattern.

Write $X=1/(\omega C)=200\Omega$. The magnitude of the series RC impedance isFor independent small errors interpreted as standard uncertainties, propagation of uncertainty givesThe input uncertainties are $u_R=2\Omega$, $u_{\omega }/\omega =0.01$, and $u_C/C=0.01$. Thus $u(Z_m{)}^2=0.80+3.20+3.20=7.20{\Omega }^2$, andThe fractional uncertainty is $1.20\%$. If the stated errors are instead worst-case bounds, their first-order contributions add in magnitude, giving approximately $4.47\Omega$, or $2.0\%$, rather than the quadrature result. Both conventions require the error interpretation to be specified.

Use the thin lens equation with positive object and real-image distances $u,v$:The transverse magnification is $M=-v/u=-f/(u-f)=-4/296$. For an object dimension $500\mathrm{mm}$, the corresponding image dimension isThe minus sign of the magnification means that the image is inverted. This is the ideal thin-lens prediction; it assumes the relevant field is admitted by the camera and ignores lens distortion.

For uniform Earth density, the shell theorem says that only mass within radius $r$ contributes to the gravitational field there. Since enclosed mass is proportional to $r^3$,At depth $h$, $g_h/g_0=1-h/R$. A simple pendulum has small-amplitude period $P=2\pi \sqrt{l/g}$, so the pendulum clock below a uniform-density surface runs at the fraction $P_0/P_h=\sqrt{1-h/R}$ of its surface-calibrated rate. After a surface-clock hour, its lag isThe mine clock is slower because gravity is weaker under the stated uniform-density assumption. The expansion $\Delta t\simeq 3600h/(2R)$ gives the same result. This is a change in the mechanical pendulum period, not a relativistic clock effect; a realistic Earth density profile would require a different interior-gravity calculation.

Use distinct symbols for the dimensions of mass, length, time, temperature and amount of substance. The molar gas constant has dimensions $\mathsf{M}{\mathsf{L}}^2{\mathsf{T}}^{-2}{\mathsf{\Theta }}^{-1}{\mathsf{N}}^{-1}$, while the given $M$ is a molar mass with dimensions $\mathsf{M}{\mathsf{N}}^{-1}$. Matching dimensions in the proposed power law giveswith the amount-of-substance equation repeating $\alpha +\gamma =0$. ThereforeDimensional analysis fixes these exponents but not a dimensionless multiplier. For example, an open-tube mode number and the gas's heat-capacity ratio enter the actual resonance frequency without changing the dimensions.

In the $e^{i\omega t}$ phasor convention, the total series RL circuit impedance is $R+i\omega L$. The resistor voltage follows the current, so its transfer function relative to the applied voltage isAt the stated condition $R=\omega L$,Thus the resistor-voltage amplitude is $\boxed{V_{R,0}=V_{\mathrm{in},0}/\sqrt{2}}$ and it lags by $45^{\circ }$. For a cosine input it is $V_R(t)=V_{\mathrm{in},0}\cos (\omega t-\pi /4)/\sqrt{2}$.