Delicate Balance

Consider a pencil that stands upright on its tip.

You might think that it would be possible to make the pencil balance for an arbitrarily long time.

It turns out that, due to Heisenberg’s uncertainty principle (which puts a constraint 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 period of time. The point is that you can’t be sure that the pencil is initially both at the tip and at rest.

Determine (roughly) the maximum time for which you can balance the pencil.

Details and Assumptions:

• Idealize the pencil as a mass of $10 \text{ g}$ sitting at the end of a massless rod of length $10 \text{ cm}.$
• $\hbar\approx 1.06\times 10^{-34} \text{ Js}.$
• $g\approx10\text{ m/s}^2.$