Harmonic Motion - Springs and Pendulums
So, now that we've talked about all sorts of stuff- linear motion, angular motion, forces, work, power, energy, even a little about gravity and space- we can focus in one a specific type of motion: motion that repeats regularly. The formal name for this kind of motion is simple harmonic motion.
I'll hold off on defining the term in words until we figure out what it looks like.
At some point in our lives, we have all played with springs, slinkies, and other similar toys. These all exhibit examples of simple harmonic motion. Imagine taking a spring and hanging it from a platform, and attaching an object to the other end:
(pardon the crudeness of the drawing- this is about the best I can do with a computer mouse.)
If you were to pull it downward and let go, you can probably guess what will happen: it will move up and down. If you were to compress the spring by pushing upward, the same sort of motion would happen. This whole section deals with how to describe this motion in much more mathematical detail than simply saying “up and down” or “left and right.”
If you've ever played with springs, they don't like being stretched; they rather prefer to stay one size. Moreover, the more you stretch and pull a spring, the harder it pushes back. As you may recall from the Forces section, Robert Hooke summarized this in the 17th Century in what he called the "law of elasticity," which we know today as Hooke's Law:
The force an object exerts to restore itself to its equilibrium is equal to its stiffness times the how far it is displaced from equilibrium . Return to the drawing:
As a general refresher, is the spring constant, and it measures how stiff a spring is. It has units of , or . The larger the spring constant, the stiffer the spring.
Also as a refresher, we can briefly recall how to determine the amount of energy contained in a spring. We can do this the "hand-wavy" way, since the "real" way is found in the energy section. If we recall the formula used for finding the work in moving an object, it is written as:
Let us assume that the force we apply to stretch or compress a spring points along the
displacement vector, and so the cosine term becomes . Ordinarily, we might just think we can express this as and be done, but the force itself varies with the displacement of the spring. So, it would be more accurate to find the "average" force between equilibrium and the final compression/stretch position, which is then expressed as:
Which lets us conclude that:
If we were to allow the mass to move back and forth, this energy gets transformed into kinetic, back to potential, back to kinetic, and so on. The total energy, however, never changes, so in general, the energy contained in a spring system is given by , where is the maximum amplitude of the oscillation, or the maximum distance from equilibrium.
Given Hooke's Law, it should be somewhat intuitive to realize that when the two forces are equal, the object is at its equilibrium position. If it is pulled away from this equilibrium and released, the net force is in favor of the spring, so it accelerates upward; as it passes through equilibrium, both gravity and the force of the spring accelerate the object back downward. That all said, we need not analyze such a complex case, just yet. Let us turn the spring sideways, such that we need concern ourselves with only one force at a time:
So, finally, the formal definition of this motion:
simple harmonic motion – motion in which the restoring force is directly proportional to the displacement from equilibrium.
This is called simple harmonic motion because we do not take into account any complicating factors, such as the mass of the spring, the shape of the object oscillating, or whether the oscillation is damped or not. We will cover such things another time.
To mathematically describe this motion, we are going to make an analogy: if we have an object undergoing uniform circular motion, and compare this motion to that of a spring, we would find that the motion of a spring is like a “projection” of an object going in a circle. Consider this drawing:
If we were to cast a light from the left side of this page across the circle, we would find that the shadow of the object undergoing circular motion would trace out a line that is the same as the mass on our spring. We can then analyze the motion this way: the force exerted on an object moving with uniform speed around a circle is described by our familiar formula:
And a projection onto the -axis would give:
Now, what is speed, in this case? It is how far around the circle we go in a given time. We can write this as:
long it takes for the object to make one full revolution; in the case of harmonic motion, it is how long it takes for one full oscillation. Substituting in, we find:
Let us examine this further: the distance along the axis is the projection that the position vector, , makes with the -axis. This means that we can write . We can also re-write the lefthand side; we know that the magnitude of the -component of this centripetal force is dependent on where around the circle the object is. This means that it is proportional to the displacement of the object from the origin. This means we can write:
There is a factor that can cancel: the displacement! This means that the rate at which our spring oscillates is independent on how far it is initially stretched. Solving for time, we find:
Which gives the formula for the period of oscillation of a spring. We can then conclude that this motion only depends on the mass attached to the spring and the stiffness of the spring. We can also write an expression for how many oscillations there are in one second by taking the inverse:
Where is the frequency of the oscillation.
Suppose we wanted to graph the position function of our spring. What would that look like? It might be somewhat intuitive that we get a sinusoidal graph, constantly varying with time. We can write:
Where is referred to as the angular frequency. For more detail on this term, see the section on rotations; for us, it is sufficient only to say that . Here, refers to the amplitude of the oscillation, or the distance from equilibrium to the maximum displacement of the spring.
We can extend this to the case pendulums, which are another example of simple harmonic motion. Let's draw what one looks like:
As you can see, gravity is the driving force of the oscillation of a pendulum, as the force of tension points along the string; tension will not be responsible for causing this object to move.. Find the component of the weight that is pulling it towards equilibrium by taking the sine of the angle:
We see here, too, the restoring force is not constant; it depends on the displacement of the pendulum bob. Let us set this as a kind of Hooke’s Law problem:
Here, refers to the arc length around the circle, which is given by - or, since is the radius of our circle, . Substituting in:
This presents us with a bit of a conundrum: if we want to have an equation that shows how the displacement angle, , changes with time, we can't isolate it in this context. But we can use a work-around: for our purposes, we can use what is known as the small angle approximation. You can read more about it in other texts, but we're just going to use it. It says, in essence, that , and it works for small angles- up to about or so. Beyond that, the error between the two expressions grows to more than , which we would not want to tolerate in our
experients. With the small-angle approximation, however, we wind up with:
And so we can find a value for our "spring constant:" . Now, if we were to substitute this into the period equation for a spring, we could find the period for a pendulum!
In more advanced topics, we can revisit this and find that if you use more complicated pendulums- like, say, swinging a stick by its end- the distance from the pivot to the center of mass of the object will follow something similar to the above relationship, though it requires more mathematics than we get into in this course. For our purposes, then, this will have to do.
We can, however, write an expression for the position of the pendulum as a function of time, which follows the same basic relationship as a spring:
Multiple Springs
Now, we enter something altogether more tricky: What happens
