Springs, Capacitors and Devices 1
Whirlwind trip through whatever things I think about. Starting with springs:
We know that when we apply a force to a spring, it stretches. We write this as follows: x = f(F) where the notation is read "extension is a function of force". Actually, what we usually do is write it the other way (this is obviously fine, after all, if the extension depends on the force applied then the force required must depend on the extension).
F = f(x) (Read as: "Force is a function of extension")
For small displacements, almost all functions are approximately linear (for the record, that's not actually good enough. Really we need experimental evidence to support saying that the extension is linear for small forces. This was discovered by Robert Hooke, as you probably know). The other condition we want to apply is that the force is in the opposite direction to the extension - this will be important later. This combines to make an overall equation:
F = -kx
(A series spring system)
This is a fairly simple looking equation, so we can hopefully explore the consequences pretty fully. Firstly, what happens when you combine springs: if they're in series then the force on both of them will be the same, and so both springs will extend by the same amount as if the other one was not there at all. What's the new spring constant then? Well, while each spring extends by the same amount as before, the system extends by their sum:
F = -(new k)(x1 + x2)
Since x1 = F / k1 (and the same for x2) then the F's cancel and we get:
1/k = 1/k1 + 1/k2 ; k = (k1 x k2)/(k1 + k2)
(A parallel spring system)
In parallel we have a similar consideration: the two springs are going to have the same extension and the force on each of them adds to the total force:
(F1 + F2) = -(new k)(x)
Since F1 = k1x, the x's cancel out and we get:
k = k1 + k2
Recognise these? They're the same equations as for resistors in parallel and in series, but the other way around (they're identical to the equations if you use conductance rather than resistance, using the equation I = CV where C is conductance).
Other interesting facts about springs, the stored energy is 1/2 k x^2 i.e. quadratic in x. This is another useful relationship, because if you have a situation with some general potential U(x), then at the minimum of that potential (which is where most things are at, remember systems tend to seek the lowest state) you can approximately describe it as a quadratic potential so long as the displacement is small. This works very well for a lot of situations, the classic example being a diatomic molecule, they vibrate pretty much like springs.
That's right! Springs oscillate! We haven't talked about that, and that's the most important part! Let's do this quickly:
(A very poor drawing of a spring oscillating back and forth. You pull the spring back, and it snaps back, gets more compressed more than it's starting position and then rebounds from the compression).
F = ma = -kx
D^2 x = -(k/m) x , where D^2 means the second derivative operator d^2/dt^2
This equation is called a differential equation (strictly speaking it's a "second order linear homogeneous" equation if you like long mathsy words). Differential equations are equations whose solutions are themselves equations. The 'order' of the equation refers to the highest derivative present (in this case, two) and also determines the number of constants in the solution.
In our case, we are going to guess that the answer is a sine wave:
x(t) = A sin(w t)
x''(t) = -w^2 A sin(w t) (differentiating twice)
Does this work? Well, yes, so long as we say that w^2 = k/m. That's great, because we now have a formula for the frequency (and in turn we can find the period. I'll leave that as an exercise.)
Well, that discussion of springs was actually longer than I thought it would be. Let's summarise quickly:
Series: 1/k = sum (1/k_i)
Parallel: k = sum (k_i) (Important! This is not true when the extension is not the same for each spring. Usually, this means the springs will be identical, but they don't actually have to be: you could have a beam suspended by three springs and only the outer two are equal. Then if you pushed in the middle, they'd all extend by the same amount.)
Potential energy: E = 1/2 k x^2
Differential equation: D^2 x = -(k/m) x
Solution for equation: x(t) = x_0 cos(wt + P)
[N.B. I've written the last equation with the constant as the amplitude (x_0), and with a 'phase lag' P in the cosine to account for all possible solutions. Normally, cosines are what you expect because you start it at maximum amplitude. But maybe you left the room and came back when it was oscillating, then you need to add some kind of phase lag for the general case.]
