In this last chapter of the course, we handle two physical phenomena which involve a linear second order constant of coefficients differential equations, say the spring mass system and the motion of the pendulum. First, let's consider the spring mass system. Attach a mass m to a spring of length l, which is suspended from a rigid support so that the spring is stretched with elongation Δl and reaches its equilibrium state. Then there are two forces acting on the mass: say the gravitational force that is the weight w = mg. m is the mass of the object and g is the gravitational acceleration which acting downward and the spring's restoring force F_s acting upward. Here, we adopt the convention that the downward direction is the positive. Here's the un-stretched spring. Then we put the mass m, then we have an elongation given by the Δl. Then by the Hooke's law, when Δ is small compared to the original length l, the restoring force F_s is proportional to the elongation. In other words, F_s = -kΔl, where the k, the positive constant k is the so-called spring constant or the stiffness of the spring. Since the mass is at its equilibrium state, we should have mg - kΔl = 0. In other words, mg is equal to the gravitational force is the same as the restoring force k times Δl. Now, let the mass be displaced further from its equilibrium state as is shown in this figure by an amount x(t), which is measured positive downward and then released. And we are going to look at the motion of this mass. Then, actually, there are four forces acting on the system: First, the weight w = mg due to the gravitation and which is acting downward and the spring force F_s = -k(Δl+x), which is acting upward when the spring is extended. In other words, the (Δl + x) is positive or the spring force is acting downward when the spring is compressed. In other words, (Δl + x) is negative. Another force that we have is the damping force F_d due to the viscosity of the fluid in which the mass is moving, which is proportional to the mass of speed so that we have F_d = -cx'(t), where the positive constant c is the damping constant and the x' is the velocity of the object. When the speed of the mass' of this is quite small. Here the negative sign, it means that the damping force is acting in the direction opposite to the mass' moving direction. Finally, there might be some external force f(t) acting on the mass. Then by the Newton's second law of motion, first we can think of the forces of the system is m times acceleration. So m times x double prime must be equal to the total force acting on the system. In other words, if we have the gravitational force mg and the spring force F_s and the damping force F_d and any external force f. That is equal to mg - k(Δl + x) - cx' + f. Since we know that at the equilibrium state mg = kΔl. So mg - kΔl cancelled out then simplifying the equation, we get mx" + cx' + kx = f. Which is a second order linear constant coefficients possibly non-homogeneous differential equation. We call this differential equation one. As the differential equation for forced damped motion for general f, if f is identically zero. Then we call it as a free damped motion. Divide the whole equation by m, divide the whole equation by m. Then you are going to get x" + cx'/m + km/x and that is equal to f/m, to give this equation. Let's set for the computational simplicity, let's call c/m is equal to two times λ and the k/m, let's say be equal to omega square and f/m let's say it to be g. Then we get the second equation say x" + 2λx' + ω square x = g(t).
