Sorry for the delay between posts. Here in Virtuosi-land, we’ve all begun our summer research projects and I think we’ve just become a little bogged down in the rush that is starting a summer research project. You feel as though you have no idea what the heck is going on, and just try desperately to keep your head up as you hit the ground running, but thats a topic for another post.
In this post I’d like to address a fun physics problem.
How long can you balance a pencil on its tip? I mean in a perfect world, how long?
No really. Think about it a second. Try and come up with an answer before your proceed.
What this question will become by the end of this post is something like the following:
Given that Quantum Mechanics exists, what is the longest time you could conceivably balance a pencil, even in principle?
I will walk you through my approach to answering this question. I think it is a good problem to illustrate how to solve non-trivial physics problems.
I will try and go into some detail about how I arrived at my solution. For most of you this will probably be quite boring, so feel free to skip ahead to the last section for some numbers and plots.
Finding an Equation of Motion
The first thing we need to do is find an equation of motion to describe our system. Lets consider the angle theta that the pencil makes with respect to the vertical. Lets treat this as a torque problem. Dealing with rotating systems is almost identical to dealing with free particles in Newtonian mechanics. Instead of Netwon’s first law, relating forces to acceleration
[ F = m ddot x ]
we just replace it with the rotational analogue of force – torque, the rotational analogue of acceleration – rotational acceleration, and the rotational analogue of mass – the moment of inertia.
[ T = I ddot theta ]
(I’ve taken the usual physics notation here, dots represent time derivatives) We need to determine the torque and moment of inertia of our pencil.
At this point I need to model the system. I need to break up the real world, rather complicated idea of a pencil, and turn it into an approximation that retains all of the important bits but enables me to actually proceed.
So, I will model a pencil as a rod, a uniform rod with a constant mass density. In doing so, I can proceed. The moment of inertia of a rod about its end is rather easy to calculate. If you are not familiar with the result I recommend you try the integral yourself.
[ I = int r^2 dm = frac{1}{3} m l^2 ]
where m is the total mass of my pencil and l is its length. I will take a pencil’s mass to be 5 g and its length to be 10 cm.
Now the torque. The only force the pencil feels is the force due to gravity, which acts from the center of mass, which for my model of a pencil occurs at half its length. I additionally wish to express the force in terms of the parameter I decided would be useful, namely theta, the angle the pencil makes with the vertical. I obtain
[ T = r times F = frac{1}{2} m g l sin theta ]
Great, putting the pieces together we obtain an equation of motion for our pencil
[ frac{1}{2} m g l sin theta = frac{1}{3} m l^2 ddot theta ]
rearranging I get this into a nicer form
[ ddot theta – frac{3}{2} frac{g}{l} sin theta = 0 ]
in fact, I’ll utilize another time honored physics trick of the trade and simplify my expression further by making up a new symbol. Since I’ve done these kinds of problems before I can make a rather intelligent replacement
[ omega^2 = frac{3}{2} frac{g}{l} ]
obtaining finally
[ ddot theta – omega^2 sin theta = 0 ]
And we’ve done it.
Looking at the equation of motion
Now that we’ve found the equation of motion, lets look at it a bit.
First off, what does an equation of motion tell us? Well, it tells us all of the physics of our system of interest. That little equation contains all of the information about how our little model pencil can move. (Notice that while I haven’t yet been explicit about it, in my model of the pencil, I also don’t allow the tip to move at all, the pencil is only able to pivot about its tip).
Great. A useful thing to do when confronting a new equation of motion is to try and find its fixed points. I.e. try and find states in which your system can be which do not evolve in time. How can I do that? Sounds complicated. In fact, I’ll sort of work backwards. I want to know the values that do not evolve in time, meaning of course that if I were to find such a solution, all of the terms that depend on time would be zero. So, if such a solution exists, for that solution the derivative term will vanish. So the solutions have to be solutions to the much simpler equation
[ sin theta = 0 ]
Which we know the solutions. In fact, lets be a little smart about things and only worry about theta = 0 and theta = pi. Thinking back to our model this suggest a pencil being straight up (theta = 0) and straight down (theta = pi).
These are the stable points of
