Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section A

Measure displacement from each arrangement's gravitational equilibrium. The series and parallel spring stiffness follows by adding forces or extensions. In the parallel arrangement the two spring forces change by $kx$ each, giving equivalent stiffness $2k$. In series the common force $F$ extends each spring by $F/k$, so total extension is $2F/k$ and the equivalent stiffness is $k/2$. HenceThe corresponding angular frequencies are $2\pi f$. gravity shifts the equilibrium extensions but does not alter the small-oscillation stiffness of these ideal linear, massless springs.

For a circular orbit, $v^2/r=GM/r^2$. The relation for circular orbital speed from surface gravity eliminates the planet's gravitational parameter: $GM=g{R}^2$. The orbital radius is $r=R+h=7.4\times 10^6$ m, soThis is about 7.4 km/s, under the circular-orbit and spherical-Earth approximation. Using surface $g$ unchanged at orbital height would overestimate the centripetal acceleration; its role here is to determine $GM$.

In the stipulated constant-speed model, the Earth-frame speed is $v=D/t=3.844\times 10^8/(76\times 3600)\simeq 1.405\times 10^3$ m/s. The Earth and Moon are stationary endpoints in this model, so their Earth-frame separation $D$ is the proper length. The spacecraft-frame separation, measured simultaneously in that frame, is $D^{\prime }=D/\gamma =D\sqrt{1-v^2/c^2}$. ThusThe spacecraft-frame distance is shorter by about 4.2 mm. This compares simultaneous endpoint measurements in two frames, not the coordinate difference between a launch event and an arrival event. The small-speed length contraction estimate also avoids numerical cancellation when evaluating the tiny contraction.

Adjacent slit contributions have phase difference $\delta =2\pi d\sin \theta /\lambda$. The six-slit amplitude is a geometric sum ${\sum }_{j=0}^5{e}^{ij\delta }=e^{5i\delta /2}\sin (3\delta )/\sin (\delta /2)$. The first zero away from the central maximum occurs at $\delta =2\pi /6$, where the denominator is nonzero. ThereforeThere is a matching zero at the negative angle. This is the first minimum of a finite slit array, rather than the much larger principal-order angle or a single-slit envelope minimum. The slits are stipulated sufficiently narrow for that envelope not to set the first zero.

Using metres and SI viscosity, the pressure difference is $\Delta P=200$ Pa and the central flow estimate isIf the quoted uncertainties are independent standard uncertainties, the pressure-difference uncertainty is ${\sigma }_{\Delta P}=\sqrt{10^2+10^2}=14.14$ Pa. First-order propagation givesHenceThe radius sensitivity of Poiseuille-flow uncertainty comes from the fourth-power dependence on radius. If the quoted errors instead mean independent worst-case bounds, the first-order absolute fractional bound is $4(0.1/2.3)+0.3/8.9+0.01/0.50+20/200=0.3276$, giving approximately $\pm 1.62\times 10^{-6}$ m³/s. That conservative bound and the standard uncertainty result are different conventions, not interchangeable errors. The supplied flow model presumes a Newtonian fluid and laminar flow that is fully developed; no density is given for an independent Reynolds-number check.

The dimensions are $[\rho ]=M{L}^{-3}$, $[\sigma ]=M{T}^{-2}$ and $[\lambda ]=L$. Equating exponents in $L{T}^{-1}=M^{\alpha +\beta }{L}^{-3\alpha +\gamma }{T}^{-2\beta }$ yieldsThus the model has $v_{\mathrm{ph}}\propto \sqrt{\sigma /(\rho \lambda )}$. Its common dimensionless factor cancels in the ratio:They are nearly equal within this capillary-wave model. The assumed power law omits gravitational restoring forces and finite-depth effects; dimensional analysis cannot establish those omissions for an arbitrary real surface wave.