Question

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.

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.

(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 acceleration 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.

Answer 1

1) The initial conditions can be written as two equations, x(0) = x_initial ... the initial position x'(0) = 0 ... the initial velocity These have the form of the function x(t) and its derivative evaluated at t = zero, so they must be equal to x(0) = C cos(ω*0) + S sin(ω*0) = C*1 + S*0 = C x'(0) = -C ω sin(ω*0) + S ω cos(ω*0) = -C*ω*0 + S*ω*1 = S*ω (the latter is because the derivatve reads x'(t) = -C ω sin(ω*t) + S ω cos(ω*t)). This gives two equations for C, S and ω: C = x_initial S*ω = 0 The thirs equation comes from the theory of simple harmonic oscillators and specifies ω in terms of the physical properties of the system, ω = √(k/m) > 0. This gives us enough information to express C, S in terms of the initial conditions, C = x_initial S = 0, or vice versa, x_initial = C, independently of S and ω, which must be fixed at 0 and √(k/m), respectively, for the sake of consistency. 2) The translation of the coordinate system reads x' = x + L This can be used to rewrite the equation of motion in the new coordinate, x'(t) = x(t) + L. Here we use the given result for x(t) to obtain x'(t) = C cos(ωt) + S sin(ωt) + L. We can also use the above results, because that's what we know about the function x(t), too. C = x_initial S = 0 ω = √(k/m) L = constant. Therefore, x'(t) = x_initial cos(√(k/m) t) + 0 + L. Hope this helps!