No. 018 October 2025.
Materials science, homework one, written with a classmate
Hooke’s law is the second term of a Taylor series
Stretch the bond. Near r0 the parabola and the bond agree; away from it they part, and the shaded gap is the error. The curve is a Morse potential, chosen for this figure. Drag the handle.
At the bottom of any smooth well the force is zero, so the first derivative vanishes and the Taylor series of the energy starts at the square term. Differentiate once more and you have a spring. Linear elasticity is this parabola, seen from far away.
U(r) ≈ U(r0) + ½ k (r − r0)²⇒F(r) = −dU/dr ≈ −k (r − r0)
what I don’t understand yet
How the stiffness k of one bond becomes the Young’s modulus a testing machine reports. My Strain tool fits E below 0.25 % strain; deriving the bridge from k to E comes next.
where this goes
In quantum mechanics this parabola is the harmonic oscillator, and its energy comes in steps: En = ħω(n + ½). That is entry 11, not written yet.