n this laboratory we will examine harmonic oscillation. We will model the motion of a mass- spring system with differential equations. Our objectives are as follows: 1. Determine the effect of parameters on the solutions of differential equations. 2. Determine the behavior of the mass-spring system from the graph of the solution. 3. Determine the effect of the parameters on the behavior of the mass-spring. The primary MATLAB command used is the ode45 function. Mass-Spring System without Damping The motion of a mass suspended to a vertical spring can be described as follows. When the spring is not loaded it has length `0 (situation (a)). When a mass m is attached to its lower end it has length ` (situation (b)). From the first principle of mechanics we then obtain mg downward weight force k(` `0) upward tension force = 0. (1) The term g measures the gravitational acceleration (g 9.8m/s2 32f t/s2). The quantity k is a spring constant measuring its stiffness. We now pull downwards on the mass by an amount y and let the mass go (situation (c)). We expect the mass to oscillate around the position y = 0