Table of Contents
1. Equilibrium, stability, and oscillationThe idea of equilibrium is a big idea from physics with applications in all branches of science. Something is in equilibrium if it is balanced out, not havingany reason to go oneway rather than another. Some places that we've seen it are:
There are also some examples you experience equilibrium in your daily life:
- 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).
As we list off these examples of equilibrium, we might start to suspect that they come in two different types.
- 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 stable equilibrium. Other examples are a ball at the bottom of a hill, a positive charge between two other positives, thermal equilibrium, electrical neutralization, and pressure.
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 oscillation. Oscillation occurs when the object has some sort of inertia that tends to carry it past the equilibrium point each time it approaches it. Oscillation is anotherbigidea in science: the molecule in a desk are oscillating when theyvibrateback and forth thousands of times per second, the temperature of the earth is oscillating when there isan ice age every ten million years. In fact, unlike most things that we've studied so far, which are sitting still or moving somewhere and then stopping, most things in your daily life tend to repeat periodically, like the cycle of classes repeating every day.
We have two ways of measuring how fast oscillation happens:
These two measures are, or course, reciprocals of each other.
- 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.
Homework: Equilibrium and Oscillation (pdf) (Due: 3/13/06)
- 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.
2. Springs are a good example of oscillationIn order to understand in more detail how oscillation works, we will study in more detail one particular oscillator that is easy to understand in terms of physics: the example of a mass hanging from a spring. We know that this is in stable equilibrium because when the mass is pulled down a little, the spring pulls more and it comes back up, and when the mass is lifted up, the spring relaxes and pushes it back down.
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 strength of the spring is described by a number called the spring constant, "k", that says how much force the spring exerts per meter of stretch.
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 by, we are talking about a change in length, from the unstretched equilibrium length to the current length. The amount of stretch is denoted by the variable "x". So, for example, if a 8 cm spring si stretched to a length of 20 cm, then x = 12 cm. If a 1 m spring is stretched to a length of 4 m, then x is 3 m.
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.
Homework: Hooke's Law (pdf) (Due: 3/14/06)
- 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.
Activity: Hooke's Law Lab (pdf) (10 pts, 3/13/06)
- 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.
Homework: Hooke's Law 2 (pdf) (Due: 3/15/06)
- 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.
3. A harmonic oscillator has the same period no matter what the size of the oscillationThe spring and mass oscillator is special because it has a particular period that it likes to oscillate with, and no matter what I do to start it oscillating, that is the only period it is willinng to have. So, for example, if I try to get it to oscillate faster by rapidly wiggling the top of the spring, the mass will refuse to play along: it will sit there and do nothing. The period that a harmonic oscillator oscillates with is sometimes called its natural period.
An oscillator that will only oscillate with a certain period is called a harmonic oscillator. A pendulum or swing is another example of a harmonic oscillator. When you start swinging on a swing, you are moving back and forth very slowly, but you have a very short distance to go; by the time you are moving faster, you are traveling a longer distance, so that the time is the same.
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:
Homework: Harmonic Oscillators 1 (pdf) (Due: 3/16/06)
- 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.
Activity: Harmonic Oscillators Lab (pdf) (10 pts, 3/15/06)
- 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.
Homework: Harmonic Oscillators 1.5 (pdf) (Due: 3/20/06)
- 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.
4. All harmonic oscillators have an acceleration that grows with the distance from equilibriumWe said before that the reason an oscillator is harmonic is that the further out you get from equilibrium, the more you are pushed back, so that when it is oscillating over a longer distance, it is also moving faster to make up for it. We can see how the acceleration (a) relates to the displacement (x) in our spring system if we plug Newton's second law, F = ma, into Hooke's law:I can do a similar thing for a pendulum by using similar triangles to find out the size of the acceleration vector. The gravity force vector, mg, and the net force vector, ma, form a triangle similar to that formed by the whole pendulum with its length L and its displacement x. So:What these equations are useful for should be evident when we look at what the equations are for the period of these two oscillators:An oscillator is harmonic if and only if we can come up with an equation that says that the acceleration is the negative of some number multiplied by the displacement. In this case, the period will be equal to 2π times the square root of the reciprocal of that number:Since some harmonic oscillators are not actual oscillating in position and acceleration, but have some other quantity that is oscillating, sometimes we make this statement more general by putting a dot over a variable to denote the rate of change of that variable with respect to time. Since velocity is the rate of change of displacement, and acceleration is the rate of change of velocity, we can write:In general, if acceleration of any variable, the rate of change of its rate of change, is equal to negative something times that variable, we have harmonic oscillation.
Homework: Harmonic Oscillators 2 (pdf) (Due: 3/21/06)
- 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".
5. Resonance can make one oscillator take energy from anotherYou know now that when you have soemthing that is a harmonic oscillator, you have to push it at its perferred frequency to get it moving (and, if there is friction, you have to keep pushing it at the same frequency). So, for example, if I am swinging on a swing, there is no sense picking a period of swinging that I particularly like and pumping at that rate; I have to "listen" to the swing and figure out what period it needs.
This is an example of an idea called resonance. The basic idea is that if I have a harmonic oscillator and I push it with an oscillation that matches its natural frequency, it will gradually swing further and further, gainign more and more energy.
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.
Homework: Resonance (pdf) (Due: 3/22/06)
- 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.
6. Oscillation Review
Homework: Oscillation Review (pdf) (Due: 3/23/06)
- 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.