Here is your problem. The magnitudes of the forces must NOT be equal, because the mass IS definitely going in one direction, it’s going up.
The whole “Magnitudes of force must be equal” thing only applies to static problems. Since there is an acceleration upward, there is a net force upward, which means that the force the rope exerts upward must exceed the force of gravity.
F = ma.
That’s all there is to it. If you can get centripetal acceleration, then you see that the net F must be a centripetal F.
I just want to say thanks for all the help getting me through this amazingly simple idea which is remarkably troubling for me.
One poster (or rather, friend of a poster) sent me via email a pretty PDF file with more explanations. The links here have been helpful, too.
However, I was thinking long and hard about this last night: the equations I give (with mgcos(theta) and such) do describe motion in a circle which is my huge problem. They just don’t describe how a friggin’ pendulum works. The paths taken are the same, the method of taking the path is drastically different.
My first problem is assuming a linear theta; that is, d(theta)/dt = 0 which it certainly doesn’t. This makes my integration all wrong to start with which complicates things entirely. I am determined to find out how a god damn pendulum works without involving centripetal acceleration explicitely. It shouldn’t be necessary to “include” that formula: the formula should be inherent in the problem.
Sorry guys, this is how I learned calculus: one theorem at a time. I have huge problems accepting things without proof, and never used theorems without proof unless required to by the teacher. I attempted (and succeeded) in proving several things myself (only got through calc up to some vector calculus and never made it into multiple variable calc). :shrug: I like to derive some things myself if I think I can; I think I can derive this, but I need to just sit down and think a little more clearly about it. Unfortunately I’ll have to assume the Period formula for now, but that’s no problem.
If I could solve a first order linear diffy-q in the general case in fucking high school this shouldn’t be beyond me. And yet sigh my impatience preceeds my abilities.
When I return it will be with generalized pendulum formulas which require no restrictions on anything except that the total angle not exceed pi radians. Until then, thanks for the strong arm of ignorance fighting!
Looks straightforward, but, because of the sin(theta) term, it’s nonlinear. In this form, the equation is basically intractable, and is solved numerically. See this web site for more info. Usually, we’re most interested in “small angles” of motion (like < 20 degrees), where sin(theta) can be approximated as theta, leaving the more easily solved linear form of the equation:
However, if you’re satisfied with the small angle approximation, you can still use the differential equation to determine the motion and then the tension in the string. This web page goes through the procedure.
erislover, for your specific problem (the maximum force the chain will ever feel), there is a solution, and Eyer8 nailed it. The maximum can be calculated for any initial angle whatsoever. If, on the other hand, you want a general formula for the force at any time, for an arbitrary angle, you’ve got problems. The answer can actually be represented exactly, in terms of the elliptic functions, but then you’ve got to find the values of the elliptic functions, so it’s basically just re-naming the problem without solving it.
I do have one nipick with what Eyer8 said: There is such a thing as centrifugal (literally, “fleeing the center”) force, depending on how you approach the problem. If you’re looking at the problem in an inertial frame, then there’s no such thing, but there is if you’re in the rotating frame.
And this simple looking but not actually simple ODE is why I’ve been saying that pretending we can use the harmonic oscillator solution for theta[sub]0[/sub] = Pi/2 is wrong. I’ve spent too much time in the last few days trying to find a better approximate solution, but alas, the best things that I’ve been able to come up with all eventually blow up, and I’ve given up because I still have to read through that anti-gravity guys (I’m pretty sure crackpot) paper and I also have to do my own research, plus homework. Fun fun. It’s truly shocking how much paper I waste every day…
You know, it’s been a long time since school, but I seem to remember you could use a Lagrangian to extract a lot of useful information from this problem.
I’ll check when I go home tonight, it has been a long time since I broke out the old mechanics books. Or maybe Chronos could tell us off the top of his head.
BTW Chronos you are right. Having not used my physics degree in eight years, it’s easier to simplify and forget.
Umm, the Lagrangian is what I’ve been using. It’s just that you still have to SOLVE the equations of motion once you get them, and there I’m stuck. Using F=ma is the worst way to go about classical physics, I think.
But, I do have more questions regarding the pendulum now. Am I correct in assuming that the velocity at any moment is a function of converted potential energy? If so, then we can determine the force of tension strictly as a function of theta (since, of course, the height is a function of theta-- namely, hcos(theta)).
…
If I ever, ever hear anyone call a pendulum “simple” again I am going to give them the what-for.
If the pendulum started at an angle alpha, then:
h = L(1 - cos(theta))
PE = mgL(1-cos(alpha)) - mgL(1-cos(theta)) = mgL(cos(theta) - cos(alpha))
KE = (1/2)mv[sup]2[/sup]
v = sqrt[2gL(cos(theta) - cos(alpha))]
and then tension = mv[sup]2[/sup]/L + mgcos(theta) = 3mgcos(theta) - 2mgcos(alpha)
Although I’d appreciate someone checking my work.
Short anecdote: I took an advanced level graduate dynamics class on computational methods of solving dynamics problems. Our first project (which spanned probably eight weeks) was to write code using several different common algorithms to solve a large-angle pendulum problem. Some of the common algorithms were suprisingly susceptable to crashing-and-burning when trying to calculate the nonlinear oscillations of the pendulum.
Looks good to me, zut. However, you have the implicit assumption that the velocity at initial angle alpha is zero. Minor point. Redefining alpha to be not the initial angle but the maximum angle would alleviate this problem, but then you wouldn’t have the formula in terms of the initial conditions, which may or may not matter to you, and is only a matter of taste.
Lagrangian dynamics is extremely useful for complicated problems, and I don’t know what I’d do without it. When you’ve only got one degree of freedom, though, it’s usually easier to use F = ma.
The mass-on-a-string is called a “simple” pendulum to distinguish it from the many possible compound pendula. For instance, you can hang two simple pendula side by side, with a spring between them, or you can hang one pendulum from the bottom of another, or allow the attachment point of the pendulum to slide around, or make the bob of the pendulum a bucket half-filled with water that can slosh around, or any of a number of other twists. Many of these compound pendula can exibit chaotic behavior, and they’re all more difficult to deal with than what we’ve got here.
zut, discourage me not! It seems like I have succeeded in avoiding the messy diffy-q on the previous page by creating a more complicated-- but more specific-- one. If we consider the initial angle to be a constant…
I’ve got a more complete formula, given two assumptions. One is that the velocity is solely a function of the kinetic/potential energy tradeoff which seems to be the case. The other is that we discuss height in terms of pendulum length exclusively. I did not want to comlicate things by picking an arbitrary height as nothing would change but the complicated-ness of the final equation.
As well, v[sub]0[/sub] = 0.
v[sub]0[/sub] = 0 is really a good assumption as even if there was an initial push the pendulum will settle out into a sweep which can be set such that
v[sub]0[/sub] = 0.
Angles are non-standard measurement such that clockwise movement is positive (simplifies equations).
a = initial angle; element of [0, 90]
b = angle moved such that 2a+t element of [0, 180]
H = initial height = L(1-sina)
h = current heigh = L(1-sin(a+b))
dh = H-h
L = length of pendulum arm
(dh/L) = sin(a+b)-sina
C = centripetal force = 2mg(sin(a+b)-sina)
N = normal component of g = mgsin(a+b)
T = C+N = 3mgsin(a+b)-2mgsina
In our case, a = h = 0, so b = 90 and T = 3mg
Because the formulas are all sine functions we can extend the domain for all measures provided we change each appearence of
sin(x)
to be
|sin(x)|
that is, so that the sine function is always positive.
To then use standard measurement then we would replace all angles with their absolute values and make their domains negative. But, as I said, this only serves to make the equations more complicated.
If we use radian measure we can see some more stuff by my calculations. I’m not going to type out the derivation, but since velocity is a function of the angle (and the starting angle, which is constant for any particular experiment) we can find the instantaneous angular velocity which I figure to be
[sup]db[/sup]/[sub]dt[/sub]= 2gv
And it seems that angular acceleration is
[sup]d[sup]2[/sup]b[/sup]/[sub]dt*dt[/sub] = 2g[sup]2[/sup]cos(a+b)
Given my track record on this thread, is there anyone who would feel like checking this?
