Musings

Pity the Freshmen

I’m teaching Freshman Mechanics this semester. Not exactly the most uplifting of subjects. Still, even here, one can find surprising tidbits of beauty.

Consider a pencil (a uniform rod) balanced on its tip. As the rod begins to fall, the tip is held in place by the force of static friction exerted by the table.

1. No matter how large the coefficient of static friction, $\mu$, the rod will reach a critical angle, $\theta$, at which it will begin to slide.
2. For small $\mu$, the tip will slide the opposite direction from the direction of fall. For large $\mu$, the tip will slide in the same direction as the direction of fall.
3. For very large $\mu$, the critical angle (measured from vertical) at which the tip begins to slide is $\cos^{-1}(1/3)\sim{70.5}$°
4. There’s a critical value of $\mu$, $\mu_\mathrm{crit}=\textstyle{\frac{15\sqrt{10}}{128}}\sim {0.37}$ at which the behaviour changes abruptly. Below $\mu_\mathrm{crit}$, the tip begins to slide opposite the direction of fall at a critical angle which approaches $\cos^{-1}(9/11)\sim {35.1}$° as we approach $\mu_\mathrm{crit}$ from below. Above $\mu_\mathrm{crit}$, the tip begins to slide in the same direction as the direction of fall at a critical angle which approaches $cos^{-1}\left( {(}48\sqrt{14}-35{)}/231\right) \sim {51.2}$° as we approach $\mu_\mathrm{crit}$ from above.

While these results sound mysterious and complicated, there is nothing but really elementary Freshman Physics at work.

That’s beautiful.