Lab IV Simple Harmonic Motion

  1. (a) Use Maple to do solve the differential equation for a simple harmonic oscillator. Let a mass, m, be attached to a spring of force constant, k. Let the initial position of the mass be at x0 and its initial velocity be v0, and obtain a solution, x(t) in terms of these constants and the angular frequency w0 = (k/m)1/2.

    2. Hints Use diff(x(t),t,t) to define the second derivative w/r to time. The second term should be omega0^2*x(t). Use sol := dsolve({eq,x(0)=x0,D(x)(0)=v0},x(t)); to solve the DFQ with the required initial conditions.

    (b) Define the kinetic and potential energy as the mechanical energy of the system and show that it is conserved.

    Hints: Write the kinetic energy term using diff(x(t),t) (after assigning the previous solution) and then add the potential energy 1/2*k*x(t)^2, then use simplify(subs(omega0=sqrt(k/m),%))

    (c) Now let x(t) = A cos(w0 t + q). Obtain expressions for A and q in terms of x0 and v0 and w0.

    Hints Define the equation for x(t) given above as equation 1. Then say eq2:= subs(t=0,eq1) and solve for A. Then substitute t = Pi/(2omega0) to get q.

    (d) Define X and V as the position and velocity functions of time, v0 and x0. Set w0 = 1 and make phase plots for the cases of v0 = 0 and x0 = 1,2 and 3.

    Hints: First unassign xo and v0 using x0 := 'x0': v0:= 'v0'. Then define the functions using unapply for x(t) and diff(x(t),t). For the phase plot, you will have plot([X(1,0,t),V(1,0,t),t = 0..2*Pi]) but its nicer to plot all on the same graph.