From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Mon, 2 Sep 2019 13:46:59 +0900 Subject: Initial commit --- macros/latex/contrib/exam-n/sample/dynamical3.tex | 59 +++++++++++++++++++++++ 1 file changed, 59 insertions(+) create mode 100644 macros/latex/contrib/exam-n/sample/dynamical3.tex (limited to 'macros/latex/contrib/exam-n/sample/dynamical3.tex') diff --git a/macros/latex/contrib/exam-n/sample/dynamical3.tex b/macros/latex/contrib/exam-n/sample/dynamical3.tex new file mode 100644 index 0000000000..abc51017ea --- /dev/null +++ b/macros/latex/contrib/exam-n/sample/dynamical3.tex @@ -0,0 +1,59 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by Declan Diver} +For a system of $N$ objects, each having mass $m_i$ and position +vector $\mathbf{R}_i$ with respect to a fixed co-ordinate system, +use the moment of inertia +\[ +I=\sum_{i=1}^N m_i R_i^2 +\] +to deduce the virial theorem in the forms +\[ +\ddot{I}=4E_k+2E_G=2E_k+2E +\] +where $E_k$ and $E_G$ are respectively the total kinetic and +gravitational potential energy, and $E$ is the total energy of +the system. +\partmarks{8} + +Given the inequality +\ifbigfont + \begin{multline*} + \left(\sum_{i=1}^N + a_i^2\right) \left(\sum_{i=1}^N b_i^2\right) \\ +\ge \left(\sum_{i=1}^N \mathbf{a}_i\cdot\mathbf{b}_i\right)^2 \\ ++ \left(\sum_{i=1}^N \mathbf{a}_i\times\mathbf{b}_i\right)^2 + \end{multline*} +\else + \begin{equation*} + \left(\sum_{i=1}^N + a_i^2\right) \left(\sum_{i=1}^N b_i^2\right) \ge \left(\sum_{i=1}^N + \mathbf{a}_i\cdot\mathbf{b}_i\right)^2 + \left(\sum_{i=1}^N + \mathbf{a}_i\times\mathbf{b}_i\right)^2 + \end{equation*} +\fi +for arbitrary vectors $\mathbf{a}_i$, $\mathbf{b}_i$, +$i=1,\ldots,N$, deduce the following relationship for the $N$-body +system +\begin{equation*} +\frac{1}{4}\dot{I}^2+J^2\le 2IE_k, +\end{equation*} +where $\mathbf{J}$ is the total angular momentum of the system. +\partmarks{8} + +Assuming the system is isolated, use the virial theorem to deduce +further the generalised Sundman inequality +\begin{equation*} +\frac{\dot{\sigma}}{\dot{\rho}}\ge 0, +\end{equation*} +in which $\rho^2=I$ and +$\displaystyle\sigma=\rho\dot{\rho}^2+\frac{J^2}{\rho}-2\rho E $. +\partmarks{8} + +Why does this inequality preclude the possibility of an +$N$-fold collision for a system with finite angular momentum? +\partmarks{6} + +\end{question} +\end{document} -- cgit v1.2.3