91Ó°ÊÓ

If the logarithmic scale were extended to include numbers down to \(0.01\), how far to the left of 1 would you have to place \(0.04 ?\)

Here is an inertia tensor (the generalization of moment of inertia) of a particular object, calculated in an ill-chosen (but Cartesian) coordinate system: $$ \left(\begin{array}{lll} 4 & 0 & 0 \\ 0 & 5 & 4 \\ 0 & 4 & 5 \end{array}\right) $$ a. Change the coordinate system to a set of principal axes, where the inertia tensor has the diagonal form $$ \left(\begin{array}{ccc} I_{x x} & 0 & 0 \\ 0 & I_{y y} & 0 \\ 0 & 0 & I_{z z} \end{array}\right) $$ and give the principal moments of inertia \(I_{\mathrm{xx}}, I_{\mathrm{yy}}\), and \(I_{\mathrm{zz}}\). Hint: Which properties of a matrix are invariant when changing coordinate systems? b. Give an example of an object with a similar inertia tensor. Rhetorical question: In which coordinate system is it easier to think of such an object? This problem was inspired by a problem on the physics written qualifying exam during my days as a PhD student. The problem required diagonalizing an inertia tensor, and there was too little time to rederive or even apply the change-of- basis formulas. Time pressure sometimes pushes one toward better solutions!

Why is the average rainfall over land lower than over the ocean?

In acoustics, sound intensity is measured by energy flux, which is measured in decibels (dB) - a logarithmic representation of watts per square meter. On the decibel scale, 0 decibels corresponds to the reference level of \(10^{-12}\) watts per square meter. Every 10 decibels (or 1 bel) represents an increase in energy flux of a factor of 10 (thus, 20 decibels represents a factor-of-100 increase in energy flux). a. How many watts per square meter is 60 decibels (the sound level of normal conversation)? b. Place the following energy fluxes on a decibel scale: \(10-9\) watts per square meter (an empty church), \(10-2\) watts per square meter (front row at an orchestra concert), and 1 watt per square meter (painfully loud).