Balancing A Pencil!

Consider a pencil that stands upright on its tip and then falls over. Let’s idealize the pencil as a mass $m$ sitting at the end of a massless rod of length $l$ .

Assume that the pencil makes an initial (small) angle ${\theta}_{0}$ with the vertical, and that its initial angular speed is ${\omega}_{0}$ . The angle will eventually become large, but while it is small (so that $\sin{\theta} ≈ \theta$ ). Determine (roughly) the maximum time the pencil can be balanced on its tip. (Assume m = 0.01 kg, and l = 0.1 m.)

NOTE : You might think that it would be possible (theoretically, at least) to make the pencil balance for an arbitrarily long time, by making the initial ${\theta}_{0}$ and ${\omega}_{0}$ sufficiently small. It turns out that due to Heisenberg’s uncertainty principle (which puts a con- straint on how well we can know the position and momentum of a particle), it is impossible to balance the pencil for more than a certain amount of time. The point is that you can’t be sure that the pencil is initially both at the top and at rest. The goal of this problem is to be quantitative about this. The time limit is sure to surprise you.