A capacitor with a capacitance $C$ is driven by a sinusoidal voltage $V(t)=V_{0}cost$ as shown in Fig. $($a$).$

$(1)$ When current $i(t)$ is written as $i(t)=i_{0}cos(t+),$ find $i_0$ and $.$

$(2)$ Find the capacitive reactance $x_C$ that is a complex number and satisfies

tilde$(V)(t)=x_C$tilde$(i)(t).$

Here the real parts of tilde$(V)(t)$ and tilde$(i)(t)$ are $V(t)=V_{0}cost$ and $i(t)=i_{0}cos(t+),$ respectively. Use $j=\sqrt[\phantom{2}]{-1}$ to avoid confusion with the current $i.$

$(3)$ Suppose that an RC circuit is driven by a sinusoidal voltage $V_{in}(t)=V_{0}cost$ and the output signal $V_{\text{o}\text{u}\text{t}}(t)=V_{1}cos(t+)$ is taken across the capacitor as shown in Fig. $($b$).$ When $R=1k$ and $C=10nF,$ find the angular frequency ${}_1$ which results in $V_1=\frac{V_0}{2}.$

$(4)$ When $={}_1,$ find $tan.$

$2.$ A point mass $m$ is attached to a spring with a spring constant $k.$ When it is subject to a damping force proportional to its velocity, the equation of motion is given by

$m{x}^{}+b{x}^{}+kx=0.$

If the point mass is initially at rest and $x(0)=x_0,$ solution to the Eq$.(2)$ is given by

$x(t)=a{e}^{-t}cos(t+),$

$1$

where $a=\frac{x_0}{cos,}=\frac{b}{2}m,=\sqrt[\phantom{2}]{{}_0^2-{}^2},tan=-\frac{}{}$ and ${}_0=\sqrt[\phantom{2}]{\frac{k}{m}}.$

Calculate the total energy of the oscillator and its time derivative to show that the energy decays at the rate of