# Quantum mechanics, balancing problem with uncertainty

1. Sep 24, 2009

### Brian-san

1. The problem statement, all variables and given/known data
Try to balance an ice pick of mass m and length l, on its point. Under ideal conditions, what is the maximum time T it can balance on its point?

2. Relevant equations
Potential Energy: U=mgh
Series Expansion for Cosine: cos(x)=1-1/2(x^2)+...

3. The attempt at a solution
By conservation of energy,
$$mgl=\frac{1}{2}ml^2\dot{\theta}^2+mglcos \theta$$
$$g=\frac{1}{2}l\dot{\theta}^2+gcos \theta$$
$$\frac{2g}{l}(1-cos \theta)=\dot{\theta}^2$$

Assume a small angle displacement, so
$$cos \theta = 1-\frac{1}{2}\theta^2$$
and
$$\dot{\theta}^2=\frac{g}{2l}\theta^2 ; \dot{\theta}=\theta \sqrt{\frac{g}{2l}}$$

Integrating from t=0 to T and theta from some small delta theta to pi/2 gives
$$T=\sqrt{\frac{2l}{g}}ln\theta\right|^{\pi/2}_{\Delta \theta}$$

The only thing I now further is
$$\Delta x\Delta p=\frac{\bar{h}}{2} ; \Delta \theta = \frac{\Delta x}{l}$$

At this point, I'm not sure how to proceed, or even if the work so far is correct. I've seen an actual solution to this, but I can't follow it and some of the math seems wrong, or is not properly explained.