- 1. Equilibrium, stability, and oscillation
- 2. Springs are a good example of oscillation
- 3. A harmonic oscillator has the same period no matter what the size of the oscillation
- 4. All harmonic oscillators have an acceleration that grows with the distance from equilibrium
- 5. Resonance can make one oscillator take energy from another
- 6. Oscillation Review

- Thermal Equilibrium: If I place a hot object near a cold one, they balance out their temperatures.
- Energy: If a ball is sitting on a flat section of track, it is in equilibrium. So, both the top of a hill and the bottom of a hill are equilibrium points.
- Electrical equilibrium: If I touch two objects together, their charge spreads out between the two. In general, everything wants to be neutral: each proton having its own electron.
- Electric fields: The zero point of a field is the equilibrium point: an object placed there doesn't feel any force.
- Forces: Wespent a lot of time trying to balance forces. A meter stick has a force of gravity acting on it, which I can balance with a tension force (holding it from above) or a normal force (balancing it on my hand).

- Pressure: If the pressure inside my ears doesn't balance the pressure outside, my ears pop. If I have a head cold, my sinuses are stuffed up and air can't get behind my ears; it's nearly impossible to equalize the pressure (so, for example, I wouldn't want to go scuba diving).

In some cases it's really easy to set something up in equilibrium and to keep itthere; if it gets pushed a little to the side it wants to swing back. A meter stick held from the top will swing back andforthpast equilibrium if pushed. In fact, the further out of equilibrium itis, the bigger the push: if I hold it really high up to start with, it swings much faster than if I move it just a little bit. We call this

In other cases, it's really hard to get in equilibrium or stay there. It's almost impossible to get a negatively charged aluminum foil ball to hang between two positive plates; if it's even a little bit off center, it will be pulled in to the nearer plate. We call this unstable equilibrium. A meter stick balanced on its end is in unstable equilibrium. (Likewise, a person standing on one foot is unstable: in football, you wouldn't want to block while standing on only one foot.)

In some of our examples of stable equilibrium, we've noticed something else: the object swings back and forth, rather than just returning immediately to the center. The physics term for swinging back and forth is

We have two ways of measuring how fast oscillation happens:

- The
**period**of oscillation, symbolized by the letter T, is how long one cycle takes. Ice ages have a long period; the wings of a mosquito beat with a short period. - The
**frequency**of oscillation, symbolized by the letter f, is how many cycles happen in one second. If two people are walking along a street, the one with shorter legs has a higher frequency: he has to take more steps in the same time to keep up.

- A situation is in equilibrium if it is balanced and unchanging.
- An equilibrium is unstable if a slight change will cause it to fall out of equilibrium.
- If a situation is in stable equilibrium,
**and**there is something (like inertia) that causes it to constantly swing**past**that equilibrium, we say that it will**oscillate**: move back and forth. - Rate of oscillation is measured with two values: the
**period**, or how long it takes to oscillate once, and the**frequency**, or how many times it oscillates in a second.

Springs follow a very simple physical law: the more the spring is stretched away from its natural length, the more force it tries to pull back with. The

So, for example, if a spring has a spring constant of 2 Newtons per meter, this means that if I stretch it by 3 m, it pulls back with a force of 6 N. Or, if I notice that it is pulling with a force of 10 N, I know that it is stretched by 5 m.

It is important to note that when we talk about how much a spring is stretched

The equation that puts all this together is called "Hooke's Law":Here, there is a negative sign just to remind you that the force pulls back in the opposite direction from how the spring is stretched. I think that, rather than using this equation, you might do better to simply remember that k is the force per meter of stretch, and just use your math common sense.

You can measure a spring constant quickly by hooking the spring with a spring scale, stretching it ouot by 10 cm along a meter stick, and then using the force to find the spring constant, the force for 1 m of stretch, using the fact that it will be 10 times the force for 10 cm.

- The more a spring stretches, the more it wants to spring back.
- The
**spring constant**of a spring says how much force it exerts per meter of stretch. - A stronger spring has a larger spring constant.

- Recognize that a spring exerts more force depending on the amount it is stretched.
- Recognize that the force increases linearly with the difference between the unstretched and stretched lengths.
- Find the spring constant of a spring.

- Review drawing force diagrams.
- Figure out what forces are acting on an object hanging from a spring, and explain why this satisfies the criteria for an oscillator: a force pulling back toward the equilibrium point, with an inertia that carries it past that point.

An oscillator that will only oscillate with a certain period is called a

As another example, if I am hopping up and down, this is an oscillation, because there is a cycle being repeated. However, it is not a harmonic oscillation, because if I jump higher, the cycles will slow down. In order for something to be a harmonic oscillator, the period must not change if I change the distance it is oscillating over.

Harmonic oscillators are interesting to us as physicists because if that oscillation is harmonic, there must be some characteristic of the oscillator itself that determines the period. In other words, if I see a pendulum hanging still or a mass and spring lying on the table, I must be able to measure soem characteristics of these objects that can be put together somehow to give an equation for the period.

So, for example, I can measure a mass in kg, and I can measure a spring constant in N/m. You should recall that N = kg m/s²; so, spring constant has units of just kg/s². If I want to put these together to get the period, in seconds, I will have to divide the mass by the spring constant and then take the square root. You will find out in the lab that we need to make a slight correction to this; the actual equation is:

- Use Hooke's law and the mass-spring oscillator period equation.
- A
**harmonic**oscillator is special because no matter how far back and forth it is oscillating, it has the same period.

- Practice using dimensional analysis to predict an equation.
- Practice using an experiment to check an equation.
- Recognize that all harmonic oscillators are governed by similar equations.

- Review equilibrium, oscillation, period, and frequency.
- Recognize that a harmonic oscillator will only oscillate at one particular frequency, no matter how I move it.
- Recognize the affect of mass and spring constant on an oscillator's period.

- Harmonic oscillation occurs when the force pushing back toward the center (the "restoring force") is proportional to the distance from the center (the "displacement").
- Any time that I can write an equation where the acceleration is something times the displacement, the period will be equal to 2π times the square root of that "something".

This is an example of an idea called

A good example of this is what happens when an earthquake hits a large city. The earthquake usually has one particular frequency that it shakes at, or some combination of a small number of different frequencies. Most buildings in the city will just shake a little bit in time with the earthquake's shaking. But if some building is unfortunate enough to have been built so that it sways in time with the frequency of the earthquake, it will rapidly start to sway more and more until it eventually collapses.

A classic example of resonance is the collapse of the Tacoma Narrows suspension bridge in 1940. You can see a long video clip of this here or a shorter version here.

I first heard about resonance in ninth grade earth science, and ever since I've seen it showing up in all sorts of strange places. It's one of those ideas that gives you a new way to think about everyday phenomena. Here's a short list of Resonance Stories.

- When a harmonic oscillator is connected to something else oscillating at that same frequency, it will also start to oscillate.
- There are many examples of this in our daily life, such as swinging on a swing set.
- This results in energy being transfered from one oscillator to the other.

- A situation is in
**equilibrim**if it is balanced and unmoving. - Equilibrium is
**stable**if pushing the situation out of equilibrium makes it swing back. - A stable equilibrium will
**oscillate**if there is some sort of inertia making it swing past the middle. - An oscillation will be
**harmonic**if the force pushing back toward the middle is proportional to the distance from the middle. - A harmonic oscillator takes the same time to oscillate once regardless of how far it is moving.