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Q.2 (a) Using Newton's 2nd Law on the standard spring-mass model, show that the equation of motion of the damped harmonic oscillator of mass M, with damping constant B and spring stiffness k, is Eq.2 M 0.0 d²x(t) dt² + B dx (t) dt + kx(t) = 0 (b) Find the general solution of Eq.2 for M = 4, B = 2 and k = 3. (c) What is the solution of Eq.2 for the special case k = 0, interpret this solution physically. (d) What is the solution of Eq.2 for the special case ß = 0, interpret this solution physically. (e) Using the expression for x(t) in part (b), find an expression for the velocity of the damped oscillator at t = 1. 1 M

(a) Using Newton's $2_{nd}$ Law on the standard spring-mass model, show that the equation of motion of the damped harmonic oscillator of mass $M$, with damping constant $?$ and spring stiffness $k$, is Eq.2 $Mdt_{2}d_{2}x(t)?+?dtdx(t)?+kx(t)=0$ (b) Find the general solution of Eq.2 for $M=4,?=2$ and $k=3$. (c) What is the solution of Eq.2 for the special case $k=0$, interpret this solution physically. (d) What is the solution of Eq.2 for the special case $?=0$, interpret this solution physically. (e) Using the expression for $x(t)$ in part (b), find an expression for the velocity of the damped oscillator at $t=1$.

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