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How long can you balance a pencil?
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 litt...
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. In this form the solution can actually most easily be written in terms of the exponential pair associated with sine and cosine, sinh and cosh (you can read more about them here, they are really neat functions). This of course means that even in principle, since our world is dominated by quantum mechanics, I could never actually balance even my model pencil forever, because I could never prepare it with perfect initial conditions.
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