SHM of Oscillating Stick due to Spring

Simple Harmonic Motion (SHM) is a periodic motion of some object caused by a restoring force that is proportional to the displacement, such as a spring.  Here is a different type of setup where a stick is lying on a frictionless table, with one end nailed to the table (which serves as the axis of rotation) and a spring attached near the other end of the stick.  If the stick is stretched slightly, so the stick forms a small angle relative to the vertical axis it had prior to stretching the spring, we can use the small angle approximation.  This says that for a small angle, the angle in radians is approximately equal to the sine of the angle.  Also, cosine of that small angle approaches 1.  We need both these approximations to solve this one!  Check it out.

Where do those Quantum Numbers come from? Using Simple Harmonic Motion to gain some insight...

In Chemistry, you learn about electron configurations, which involves learning the rules for 4 quantum numbers. Three of these numbers are integers. But students tend to be mystified by where these suddenly and almost magically appear. Why integers? Why the values that you are forced to memorize?

It all starts with the heart and soul of quantum mechanics, which is the Schrodinger equation. This is the F = ma of quantum land. For this case, our system is an electron oscillating back and forth between two walls. It is a nice, neat 1-D system. When we see what the Schrodinger equation looks like in this case, it will be identical to what we get for a mass on a spring oscillating back and forth. Since we know the solution of simple harmonic motion is a sine or cosine, then the solution of what turns out to be the wave function for our electron is also a sine or cosine. We will see that the electron will be restricted in its energy, that it will have restricted energies determined by an integer that we get as part of our solution! It is a quantum number!

These quantum numbers are simply part of solutions to complicated equations you get from the Schrodinger equation. An exact solution exists for a hydrogen atom, for example. You get three integers for an electron in an atomic orbital because it is a 3-D system rather than the 1-D system we deal with here. But the idea is the same. Integers are natural parts of these solutions, and they then are part of energy solutions for atoms and particles, which tells us that the energies of the electron have specific, allowed values, and not a continuum of energy that we see for a superball bouncing between two walls in a big, macro-world. The micro-world follows a separate set of rules in quantum mechanics. I hope this helps.

By the way, for an example of how the quantum numbers play out for the periodic table, check out some rules of the game.

How to get Simple Harmonic Motion Solution for a Pendulum

A pendulum is technically NOT simple harmonic motion like a spring. SHM is defined when a force is proportional to the displacement of the object, just like a spring has F proportional to x. A pendulum is close, but the restoring force, being the tangential component of gravity, is proportional to sin(theta). Check out this video to see what we mean by the small angle approximation, or sin(theta) ~ theta (in radians) when theta is small, or about 10-degrees or smaller. This approximation works well, and you should check it on your calculator to prove it to yourself if you are not familiar with this. So for small angles, a pendulum is mathematically the same as an oscillating spring, and therefore is SHM and has a known solution of sine or cosine of wt plus a phase angle.

How to solve simple harmonic motion problems using initial conditions

We have seen how the general solution for simple harmonic motion involves sines or cosines. These functions are periodic functions, and it makes sense that they are used to solve period motion problems. But what about problems where we are given initial conditions (i.e. at t = 0)? How do we find specific solutions to specific problems? Well, here is an example of how to do this. We will have a case where a mass oscillating on a spring has BOTH initial position and initial velocity. We introduce the general solution that involves a phase angle. Check it out.

How to find solutions for simple harmonic motion (SHM)

Simple harmonic motion is a subset of periodic motion, defined as motion that depends on displacement. Springs certainly follow this definition since Hooke's law gives us F = -kx. This picture of a mass oscillating at the end of a spring also makes use of energy, U = .5kx^2. But our goal is to understand the motion as a function of time, as well as a function of position. Bringing time into the picture is the issue, and it turns out we need a second order differential equation. This video outlines how to get a solution for position of the mass on a spring as functions of time. Turns out these require sine or cosine functions. Other videos will focus on other details and specific problems to show a general solution we can use.