PH-UY 1013 Lecture Notes - Lecture 12: Tl;Dr, Damping Ratio, Quadratic Equation
Document Summary
Given a pendulum of mass m with a linear damping force (force increases with velocity). The damping force has a constant of c and the spring force has constant k. using these constants, we can create a force equation using newton"s 2nd law. = m d2x dt2 + b dx dt. This second order ode has a solution in the form of x = ert. Substituting this into the equation and factoring out e t will give the characteristic polyno- mial: mr2 + br + k = 0. From there, the quadratic equation is used to nd up to 2 possible solutions. r = b b2 4mk. From here, there are 3 di erent cases: underdamped (b2 4mk < 0), overdamped (b2 . 4mk > 0) and critical damped (b2 4mk = 0). For the purposes of physics c, we will only be dealing with underdamped oscillators. Underdamped oscillators reach the zero position the fastest, but then oscillate around it.
