phy53 lect dr. brown 16 november 2010
curve balls -- baseball
discussion -- golf balls...dimpled surface; compare to ping pong ball
leading to Young's Modulus
atoms modeled like connected by springs
[discussion of similarity -- Leonard-Jones potential -- consider Taylor series expansion about the minimum]
single chain; two chains; block of material with cross sectional area...
F_{restoring} = - Y A/L \Delta L = -"k_eff" \Delta L
Y = young's modulus = stress over strain = (F/A)/(\Delta L /L)
explains linear response limit, elastic limit[permanent deformation] and fracture limit
Another material response -- push rectanglular cross section at top -- shear stress and strain
Shear modulus = shear stress / shear strain = (F/A) / (\Delta x /L)
finally -- "Bulk modulus" increase pressure -- response is change in volume
large bulk modulus => low compressibility
We can really understand springs now and where, from microscopic view of matter, Hooke's Law comes from-- so,
solve -- mass on a spring
F_x = -kx = m a = m \frac{d^2 x}{dt}
Simple Harmonic Oscillator Equation
\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0
solve it Euler equation...real part ... x(t) = x_o cos (\omega t + \delta)
next -- pendulum ... nonlinear diffy q so, make small angle approximation sin(\theta) approx \theta