You considered rotational energies of things like oxygen molecules and elec- trons around such molecules. you worked out some typical energy results for rota- tional kinetic energies of such objects. of course, a complete treatment requires that we use quantum mechanics, but you have seen that classical mechanics has a great deal to say about how such systems work, particularly if you insert some appropriately chosen factors of planck's constant at just the right place. many of these "classical- calculations-with-an-ħ" are quick routes to approximate results and insight into the behavior of quantum systems, and are used regularly by working physicists. in this part of the course, we are considering systems of coupled masses and springs. let's continue down the path of using classical calculations along with some quantum insight to learn something about the behavior of molecules. on hw13, you considered an oxygen molecule that consists of two oxygen nuclei, each of mass m = 2.7 x 10-26 kg and separated by a distance 1.2 x 10-10 m. let's change things up a bit: consider a molecule of carbon monoxide. what are the two atomic masses? what is the quoted bond length you can find online (a google search will do the job)? this distance must represent an equilibrium position of one atom from the other and certainly in the modern view, calculating it would require quantum mechanics. from hw13, you know that the electronic kinetic energy is typically much higher than that of the motion of the atomic center of mass or rotational motion. your chemistry courses have also taught you that the electrons are shared in some way between the atoms. electrostatic energies play an important role. again, quantum mechanics is needed to get it all right. thus, you can understand why replacing all of that with a moderately accurate spring would be a huge simplification. (a) what might be an order of magnitude estimate for the value, k, for the 'spring' between the carbon and oxygen atoms of a co molecule? anything short of a full quantum calculation is just a guess, but you can use some rough numbers to get a rough estimate: a typical chemical bond energy (the decrease in energy between the bonded atoms in a molecule compared to the energy of the two isolated atoms) is measured in electron-volts. the co molecular bond is actually the strongest known of the dipolar molecules and is close to 10 ev. we are going to approximate the bond behavior as being like a spring, but if you pull the spring apart enough to exceed the chemical bond energy, the spring is going to break. let's assume that pulling the atoms to twice the equilibrium distance is enough to break the bond. set the ideal spring energy for a stretch of twice the equilibrium bond length to equal the bond energy and estimate the numerical value of the co spring constant in units of newtons/meter. (b) let's simplify the problem by considering only 1-d motion for the carbon and oxygen atoms. write down the 1-d lagrangian in terms of the coordinates and velocities of the carbon atom and the oxygen atom. (c) now do a coordinate transformation to change from the carbon and oxygen co- ordinates to center of mass and relative coordinates. write the lagrangian in these coordinates. name at least one conserved quantity that you can identify from this lagrangian. explain how you know that the quantity is conserved.(d) now consider the relative coordinate part of the lagrangian. write down the equation of motion for the relative separation. identify the 'natural' frequency of oscillation for the relative motion of the co molecule. estimate the frequeny of relative oscillation given that atomic masses and your estimate of the spring constant. (e) electricity and magnetism and quantum mechanics both tell us that the co molecule will respond to electromagentic waves that are at frequencies close to the natural frequency of the relative mechanical oscillation. given your result above, in what part of the electromagnetic spectrum do you expect to find electromag- netic waves that interact strongly with the vibrational modes of dipole molecules like co? x-ray? ultra-violet? visible light? infrared light? microwaves? radio waves? higher frequencies? lower?