(Solved): - Consider a freely oscillating mass-spring system (The simplest model that captures the essential ...

- Consider a freely oscillating mass-spring system (The simplest model that captures the essential features of a free mechanical vibration) as illustrated below: We can model the motion of the spring mass system using the OOE: $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$ where $x(t)$ : The distance to the mass from the equilibrium position $(x=0)$ of the spring mi mass, ci damping constant, $k:$ spring constant A motion is said to be oscillatory if the displacement $(x(t)$ in this case) varies according to a sinusoidal function of the form: $\begin{array}{l}x(t)=C\cos (\omega t)+D\sin (\omega t)=A\sin (\omega t+\varphi )----(2)\\ \text{ where }A=\sqrt{C^2+D^2}\text{ and }\tan \phi =\frac{C}{D}\end{array}$ in which A: The amplitude-maximum displacement from the equilbrium position (could depend on time) e: Angular frequency $\Phi$ : Phase constant Further. Period of the oscillation - The time taken to complete ose cycle in the motion: $T=\frac{2\pi }{6}$ Frequency of the oscillation - The number of cycles completed per unit time: $f=\frac{1}{T}=\frac{e}{2\mathrm{g}}$ Phase shift - The horizantal shift of the function from the original position: $\varphi =\frac{\phi }{\omega }$ 1. For a mass-spring motion as described by the $ODE$ above, $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$ oscillatory motion occurs only if Case 1: Damping constant, $c=0$ (no damping - Simple Harmonic Motion) or Case II: $c^2-4mk<0$ (under damped) Find the solution of the OOE (1) under case I and case II, expressing the solutions in the form (2) and then find the amplitude, frequency and phase shift in each case. 1. For a mass-spring motion as described by the ODE above, $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$ oscillatory motion occurs only if Case I: Damping constant, $c=0$ (no damping - Simple Harmonic Motion) or Case II: $c^2-4mk<0$ (under damped) Find the solution of the ODE (1) under case I and case II, expressing the solutions in the form (2) and then find the amplitude, frequency and phase shift in each case. 2. Assume a mass-spring motion is described by the ODE, $\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+\frac{9}{4}x=0,x(0)=1,x^{\prime }(0)=0$ which describes the motion of a mass of $1\mathrm{kg}$ attached to a spring with spring constant $\frac{9}{4}{\mathrm{Nm}}^{-1}$ moving under a damping force of $b(N-\sec /\mathrm{m})$, which was initially displaced $1\mathrm{m}$ (to the positive direction) from the resting position and let go (i.e. with initial velocity 0 ). Find the solution of the system when i. $b=0$ ii. $b=1$ iii. $b=3$ iv. $b=5$ Qualitatively sketch the graph of $x(t)$ for each instance. For instances when the motion is oscillatory, find the amplitude and the frequency of the motion. [you may use the expressions from question 1] $\$$ Please refer to section 4.9 in the text (page 212 - page 219) for a detailed explanation of the situation and for examples.