To understand the application of the general harmonic equation to finding the acceleration of a spring oscillator as a function of time. One end of a spring with spring constant k is attached to the wall. The other end is attached to a block of mass m. The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be x = 0. The length of the relaxed spring is L, shown in the Fig. 3.
The block is slowly pulled from its equilibrium position to some position xinit > 0 along the x axis. At time t = 0 , the block is released with zero initial velocity. The goal of this problem is to determine the acceleration of the block a(t) as a function of time in terms of
Fig. 3: Figure for problem 6
k, m, and xinit. It is known that a general solution for the position of a harmonic oscillator is x(t) = C cos(ωt) + S sin(ωt), where C, S, and ω are constants (Fig. 4). Your task, therefore, is to determine the values of C, S, and ω in terms of k, m,and xinit and then use the connection between x(t) and a(t) to find the acceleration.
Fig. 4: Figure for problem 6
(a) Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block a(t) as a function of time. Express answer in terms of k, m, and x(t).
(b) Using the fact that acceleration is the second derivative of position, find the acceler- ation of the block a(t) as a function of time. Express your answer in terms of ω and x(t).
(c) Using your solutions from (a) and (b) find the angular frequency ω. Express your answer in terms of k and m.