 We're going to continue through basic spectroscopic physics by looking at simple harmonic motion. Specifically, we're going to look at reduced mass, which is useful for generating analytical solutions to two body systems that are either rotating or vibrating against each other. Now, an analytical solution means that we can create a perfect prediction of where a particle will be at any point in time. We'll generate some function of time that spits out their properties at any time we choose. These analytical solutions are in direct contrast to numerical solutions, which are approximate and unpredictable. Now, unpredictable doesn't mean completely random or non-deterministic, but it does mean that we can't get that perfect solution that gives us the location of a particle at any point in time. To do the calculations here, we have to run a simulation and then any estimates or approximations we make can generate chaotic results further down the line. Usually, the typical example here is the double pendulum. Absolutely tiny, one in a thousand changes can diverge rapidly after a certain time and that's all unpredictable. Thankfully, we're not going to cover those because that's what computers are for. So, let's look at doing an analytical solution for a very simple system, a single spring. If there are two masses on the end of a spring, they will move and bounce against each other and their movements will depend only on their mass and how far they stretch or compress the spring. First, we need to build a dynamic equation to describe how they're going to move. Now, a dynamic equation is something that's based on the second law of motion or F equals MA or force equals mass times acceleration. If we know the force acting on an object and we know it's mass, we know it's acceleration, i.e., where it's going. So, what is the force here? For a spring, the force applied to these masses is directly proportional to the displacement of the spring from its equilibrium or resting length. Now, that gives us something to put on the other side of that equation. The displacement, which will label X multiplied by a proportionality constant K. Now, this model tells us the restoring force is directly proportional to displacement and the resulting acceleration or change in velocity can be determined from that displacement only. Now, the spring here is just metaphorical. This fact about restoring forces and displacements is an approximation for a lot of other forces and systems, including how two atoms will vibrate against each other with a force generated by electrostatic attraction and repulsion instead of a metal spring. Anyway, regardless of where the forces come from, we have two masses and we have two dynamic equations. The respective forces are in opposite directions, one positive and the other negative. And let's also go ahead and change A for acceleration to some second derivatives of the position of each of the two particles labelled 1 and 2 to help tell them apart to make it look like, well, like proper maths. Now, we've got two simultaneous equations. They're happening at once, so we should be able to combine them. But not immediately, because we can't get anything useful just by equating them. That just tells us the forces are equal, but opposite again. So first we'll multiply each one by the opposite equations, maths, and then we'll subtract the two, get them into one same equation. Now, there are a few bits here that can be factored out. And when we rearrange all of that, we get a new equation very similar to one of the original dynamic equations. There's a single expression for force. That's the spring constant K and the displacement X. And there's an acceleration, which is now in terms of how the total length between the two objects changes. And that acceleration multiplied by mass, or at least something that has the same dimensions of mass, if you follow through with the units. The value here is known as the reduced mass, and it's called this because it reduces a two-body problem to a one-body problem. Now, how does that actually work? Well, if the reduced mass is a single mass here, the Greek letter mu, this is the mass a single object would have if it was vibrating with the exact same frequency and displacement, but against a completely fixed point. So we have a one-body problem and one dynamic equation that we can follow. For this system, the time-dependent solution to this equation needs to harmonic motion. It's a simple sine or cosine oscillation from one extreme to the other. Now, with a real physical spring, that motion is eventually lost to friction, but in systems where the friction isn't an issue, but the force is still proportional to displacement, such as, say, a chemical bond, that motion is continuous. Now, one important property here is that the oscillation happens at the same frequency, no matter the displacement. When we consider the quantum case further down the road, that frequency is the same, regardless of the energy level. The next reduced mass case we're going to look at is rotational or orbital motion. So, with two objects all orbiting each other, such as electrons around an atom, a planet around a sun or two atoms, in a diatomic molecule that are rotating, they actually orbit a common center of mass. That again gives us two dynamic equations, one for each object. Now, I'm not going to cover the derivation of reduced mass in this case, but we can substitute in reduced mass using the exact same formula as it before, and describe the rotation, in this case, as an object with reduced mass orbiting a fixed point. This would have the exact same properties as the two body systems. So, that includes the rate or the speed of rotation and the distance between the objects. Finally, a few properties of the reduced mass to get an intuition of what it does. If the masses are similar, then the reduced mass tends towards half their average mass value, or the other end of the scale and the other extreme. If the masses are very different, the reduced masses tends towards the value of the smallest one. Now, this latter case makes a lot of sense if you consider systems that have a very big difference in their mass. So, for instance, the Earth appears to orbit the Sun because it's so much smaller. The reduced mass of that system is virtually the same as the Earth's mass within any reasonable error. So, the systems actually look the same. Even for a much larger system, such as the Sun-duper reduced mass, the difference between Jupiter's mass and the reduced mass of the system is about one part in a thousand. So, that's the power of the reduced mass. We can simplify a two-body system to one and create a perfect, predictable solution. Now, for three or more bodies, it's a lot more complicated and fortunately chaotic, but thankfully we won't be going into that in too much detail at all. It's a bit complicated and yet that is what the computers are for.