Wednesday, July 23, 2008

Answer: Inverted Pendulum 2
New Problem: Guitar Frets

Inverted Pendulum 2
The longer the inverted pendulum, the easier it is to balance. The braindead way to see this is to notice that the torque on the pendulum is proportional to its length, while the moment of inertia is proportional to the square of the length. So the angular acceleration is inverse-proportional to length.

The period is longer, for the same reason. This difference between the pendulum oscillating through small angles with the weight near the bottom, and falling through a small angle with the pendulum near the top, is that the sine function becomes a hyperbolic sine. The time constant is unaffected.

If you have two pendulums, one with a longer arm but both with the same mass, and raise them to the same angle and let them go, the tension in the pendulum arms will be the same magnitude. The pendulums are going in circles, so their acceleration is v2/r. "r" is the length of the pendulum. "v2" is proportional to the kinetic energy. The kinetic energy comes from converting potential energy, and the potential energy is linear in the length of the pendulum. So the v2 term is also linear is the length. Dividing them, the acceleration and force are independent of the length.

Guitar

Why do the frets on a guitar get closer together as you move down the neck?

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