PHYS 211 Lecture 5 - Oscillations: Simple Harmonic Motion 5 - 1
PHYS 211 Lecture 5 - Oscillations: simple harmonic motion 5 - 1 Lecture 5 - Oscillations: simple harmonic motion Text: Fowles and Cassiday, Chap. 3 Consider a power series expansion for the potential energy of an object 2 V(x) = Vo + V1x + V2x + ..., where x is the displacement of an object from its equilibrium position, and Vi, i = 0,1,2... are fixed coefficients. The leading order term in this series is unimportant to the dynamics of the object, since F = -dV/dx and the derivative of a constant vanishes. Further, if we require the equilibrium position x = 0 to be a true minimum in the energy, then V1 = 0. This doesn’t mean that potentials with odd powers in x must vanish, but just says that their minimum is not at x = 0. Thus, the simplest function that one can write is quadratic: 2 V = V2x [A slightly more complicated variant is |x| = (x2)1/2, which is not smooth at x = 0]. For small excursions from equilibrium, the quadratic term is often the leading-order piece of more complex functions such as the Morse or Lennard-Jones potentials. Quadratic potentials correspond to the familiar Hooke’s Law of ideal springs V(x) = kx 2/2 => F(x) = -dV/dx = -(2/2)kx = -kx, where k is the spring constant. Objects subject to Hooke’s Law exhibit oscillatory motion, as can be seen by solving the differential equation arising from Newton’s 2nd law: F = ma => -kx = m(d 2x/dt 2) or d 2x /dt 2 + (k/m)x = 0. We solved this equation in PHYS 120: x(t) = A sin( ot + o) Other functional forms such as cosine can be changed into this form using a suitable choice of the phase angle o.
[Show full text]