Chaos ~ another limit of classical theory Near points of instability the expected predictability of classical law can break down. This happens because of our necessarily incomplete knowledge of initial conditions. The new feature is that almost identical initial conditions lead to completely different outcomes. Example: pencil stands on sharp point : metal near non-uniform magnetic field

Chaos

Poincare Plots

Brownian motion – an example of fractal paths Over a vast range of time steps Brownian motion is fractal so that a precise velocity cannot be assigned. If we call the path length L and the step size x then they are related via a power law: L = const. * x(1-D) where D is the ‘fractal dimension’ of the path A normal, smooth curve has L independent of x, or D = 1. For Brownian paths one finds D = 2. As we make x smaller, L increases. The molecular collison time (mean free path time) sets a limit.

Attractors

Magnetic brakes

Gravitational Torsional Balance - Cavendish Experiment weighing the earth A principal discussion of the torsion pendulum following two methods. Manual procedures differ a little.

What we measure:

so neglect (d/w)2 Is max at b = 51.6 degree so b’ =0 and

Corrections: Issue other masses nearby Periods T1, T2 show fluctuations ~ 1ms Issue time fluctuations

Issue sphericity For accurate results DT should be as large as possible