Past exam of the physics B course of the University of Cambridge / 2013 / Paper 1 / Section A

For hydrostatic volume strain from Young's modulus and Poisson's ratio in isotropic linear elasticity, the bulk modulus is $K=E/[3(1-2\nu )]=5.56\mathrm{GPa}$. Hydrostatic compression therefore givesHere $p=101325\mathrm{Pa}$ is the pressure increase, and the negative sign denotes contraction. Equivalently each normal strain is $-(1-2\nu )p/E$ and the three strains add. This is the small-strain isotropic model; concrete's microstructure is represented only through the supplied elastic constants.

Put line charges $+\lambda$ and $-\lambda$ on the two axes. For $d\gg a_1,a_2$, the surface potentials differ byThus the capacitance per unit length in vacuum isThe two logarithms belong to one potential difference, not two independent capacitances in parallel. A homogeneous dielectric replaces ${ϵ}_0$ by its absolute permittivity; finite radii and proximity corrections are neglected.

With the standard liquid-water approximation $\rho \simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$, each portion has mass $100\mathrm{kg}$. The equal heat capacities give final temperature $T_f=310\mathrm{K}$ by energy conservation. For an insulated mixing process with no other changes,The reversible integrals $mc\int dT/T$ evaluate entropy differences between equilibrium states; the actual heat transfer is irreversible. These are two portions of the same substance, so there is no additional ideal-mixture compositional entropy.

Let absorbed solar flux per unit surface area be $F_{\odot }=\sigma T_0^4$. The single-layer greenhouse model emits infrared upward and downward. Top-of-atmosphere balance gives $\sigma T_a^4=F_{\odot }$, hence $T_a=T_0$. Surface balance is $\sigma T_s^4=F_{\odot }+\sigma T_a^4$, soThe layer simultaneously satisfies $\sigma T_s^4=2\sigma T_a^4$. This ideal radiative model excludes convection, spectral windows and atmospheric temperature structure; it is not a quantitative prediction for the real atmosphere.

For the tangent-axis inertia of a spherical shell, subtract the missing solid core from a solid sphere:The parallel axis theorem shifts the axis a distance $R_2$:It tends to $7M{R}_2^2/5$ for a solid sphere and $5M{R}_2^2/3$ for a thin spherical shell.