\f1. From the graph in Fig. 1 answer the following questions. (a) Which curve represents y = y(t)? How do you know? Blue curve (b) What is the period of the motion? Answer this question first graphically (by reading the period from the graph) and then analytically (by finding the period using wo). (c) We say that the mass comes to rest if, after a certain time, the position of the mass remains within an arbitrary small distance from the equilibrium position. Will the mass ever come to rest? Why? (d) What is the amplitude of the oscillations for y? O.8 meters (e) What is the maximum velocity (in magnitude) attained by the mass, and when is it attained? Make sure you give all the t-values at which the velocity is maximal and the corresponding maximum value. The t-values can be determined by magnifying the MATLAB figure using the magnify button , and by using the periodicity of the velocity function. How far is the mass from the equilibrium when the maximum velocity is attained? How does the size of the mass m and the stiffness k of the spring affect the motion? Support your answer first with a theoretical analysis on how wo (and therefore the period of the oscillation) is related to m and k, and then graphically by running LAB05ex1.m first with m = 4 and k = 4, and then with m = 1 and k = 10 . Include the two corresponding graphs. Need help with Parts B,E & FEquation (2) is rewritten ([2 y F+w3y=0 (3) where (4:3 = k/m. Equation (3) models simple harmonic motion. A numerical solution in the interval 0 S t g 10 with initial conditions y(0) : 70.8 meter and 5&0) : 0.3 . (i.e. the 111355 is initially compressed upward 0.8 meters and released with an initial downward velocity of 30cm/s; see setting (c) in figure), m : 1, .E: = 4, is obtained by rst reducing the ODE to rst-order ODES (see Laboratory 4). Let U : 3/. Then v' : y\" : iwy : 743;. Also. *u(0) : y'(0) : 0.3. The MATLAB program LABOSexl implements the problem. In 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 massspring system from the graph of the solution. 3. Determine the effect of the parameters on the behavior of the inassrspring. The primary MATLAB command used is the ode45 function. MassSpring 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 En (situation (a)). When a mass m is attached to its lower end it has length If (situation (b)). Rom the first principle of mechanics we then obtain m + 7k E 7 E : (J. 1 9 ( u} ( ) downward weight force upward tension force The term g measures the gravitational acceleration (g z 9.8m/32 z 32 f t. / 9'!) The quantity k is a spring constant measuring its stiffness. \"'13 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 1; = U. The second principle of mechanics yields 2 2 mg { k( I 3,: En) m d (IEH , i.e., md 2?} +ky = U (2) V \\/ [1152 (if.2 weight upward tension force . acceleration of mass using (1). This is a secondrorder ODE. TH (a) (b) (C) ((1')