Introduction to Classical Mechanics · Problem 3.13

Using F = ma — Morin

Physics — Mechanics · balancing_pencil

Problem

Balancing a pencil *** Consider a pencil that stands upright on its tip and then falls over. Let's idealize the pencil as a mass mm sitting at the end of a massless rod of length ℓ\ell. (a) Assume that the pencil makes an initial (small) angle θ0\theta_0 with the vertical, and that its initial angular speed is ω0\omega_0. The angle will eventually become large, but while it is small (so that sin⁡θ≈θ\sin \theta \approx \theta), what is θ\theta as a function of time? (b) You might think that it should be possible (theoretically, at least) to make the pencil balance for an arbitrarily long time, by making the initial θ0\theta_0 and ω0\omega_0 sufficiently small. However, 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 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. Without getting into quantum mechanics, let's just say that the uncertainty principle says (up to factors of order 1) that ΔxΔp≥ℏ\Delta x \Delta p \geq \hbar, where ℏ=1.05⋅10−34\hbar = 1.05 \cdot 10^{-34} J s is Planck's constant. The implications of this are somewhat vague, but we'll just take it to mean that the initial conditions satisfy (ℓθ0)(mℓω0)≥ℏ(\ell \theta_0)(m\ell \omega_0) \geq \hbar. With this constraint, your task is to find the maximum time it can take your θ(t)\theta(t) solution in part (a) to become of order 1. In other words, determine (roughly) the maximum time the pencil can balance. Assume m=0.01m = 0.01 kg, and ℓ=0.1\ell = 0.1 m.

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