The discussion revolves around the use of Taylor expansion to express potential energy, particularly in the context of spring potential energy as described by Hooke's Law. Participants explore the theoretical underpinnings and practical implications of using Taylor series in this context.
Participants generally agree on the utility of Taylor expansion in expressing potential energy, particularly in the context of small displacements in springs. However, the initial question regarding the necessity of Taylor expansion versus direct application of Hooke's Law remains open for further exploration.
The discussion does not resolve whether Taylor expansion is strictly necessary in all cases or if Hooke's Law can suffice in certain scenarios, indicating a potential area for further inquiry.
Thank you that explained it in great detail!Chandra Prayaga said:The Hooke's law force, F = -Kx, defines an ideal spring. In a real spring, Hooke's law is valid only for small stretches (small x) of the spring. The Taylor expansion of the potential energy about the equilibrium position (in this case, x = 0), is used to express more general potential energies, not necessaily of an ideal spring. Since any function (inifinitely differentiable) can be expanded in a Taylor series, even the potential energy can be expanded in a Taylor series. If you are expanding the potential energy about its minimum (at x = 0), then
U(x) = U(0) + x (dU/dx)0 + (x2 / 2!) (d2U/dx2)0 + ...(higher order terms)
where the subscript 0 on the derivatives means that they are evaluated at x = 0. If x = 0 is a minimum, then the first derivative in the above expansion is zero, and that leaves the quadratic term as the dominant term in the Taylor expansion of the potential energy, and is identified as the "spring" potential energy.