Investigation of Simple Harmonic Motion There are two common forms of the equations used to model simple harmonic motion (SHM), which is the motion of springs, swings, tides, and many other periodic phenomena. These equations are y (t) = Asin(wt + $) and y (t) = casin wt + cicos wt, where: y(t) = distance of weight from equilibrium position w = angular frequency (measured in radians per second) A = amplitude $ = phase (depends on initial conditions) C1 = Asing C2 = Acoso Suppose you are an engineer trying to recreate an experiment involving a weight on the end of a spring. This simulation C will give you an idea of what the experiment will look like. For more information, you can visit this simple harmonic motion C website. You are given the equation y (t) = 2 sin 4at + 5 cos 4at, which models the position of the weight, with respect to time. You need to find the amplitude of the oscillation, the angular frequency, and the initial conditions of the motion. You will also be required to find the time(s) at which the weight is at a particular position. To find this information, you need to convert the equation to the first form, y (t) = Asin(wt + $). Part A Use the information above and the trigonometric identities to prove that A sin(wt + $) = casin wt + cicos wt. BIU x X 2 15px vPart B To rewrite y (t) = 2 sin 4at + 5 cos 4xt in the form y (t) = A sin(wt + $), you must first find the amplitude, A. Use the given values C1 = A sing and c2 = A coso, along with the Pythagorean identity, to solve for A. BIU X X 2 15px v AvLV E V BV Characters used: 0 / 15000 Part C To rewrite y (t) = 2 sin 4at + 5 cos 4xt in the form y (t) = A sin(wt + $), solve for p. Show your work and solution in the response box. BIU X X 2 15px A v V Characters used: 0 / 15000Part D Write y (t) = 2 sin 4xt + 5 cos 4xt in the form y (t) = A sin(wt + ) and identify the amplitude, angular frequency, and the phase shift of the spring motion. Record your answers in the response box. BIU X X 2 15px V AvLv VEV Characters used: 0 / 15000 Part E The angle $ represents the phase shift, determined by the initial conditions of the experiment or the position of the weight at t = 0. If the weight is at its maximum positive position (weight is above equilibrium) at t = 0, then $ = 0. If the weight is at its maximum negative position (spring is stretched and weight is below equilibrium) at t = 0, then $ = 7. If the weight is traveling in the negative direction and passing through equilibrium at t = 0, then o = . In the response box, describe the initial condition of our experiment; specifically, describe the position of the weight and the direction in which it was traveling. BIU X X 2 15px AVIVPart F Find the times (to the nearest hundredth of a second) that the weight is halfway to its maximum negative position over the interval 0