Phasor circuit problem: 1 resistor, 1 capacitor, 1 inductor, 1 Voltage source

[VinnyCee]

Homework Statement

What value of $\omega$ will cause the forced response $v_0$ in the circuit below to be zero?

http://img248.imageshack.us/img248/1854/problem934ao5.jpg

Homework Equations

Phasor eqs.

The Attempt at a Solution

$$i\,=\,\frac{50\,-\,V_1}{2\Omega}\,=\,\frac{V_1\,-\,V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}$$

If I substitute $V_1\,=\,0$, then I get:

$$25\,=\,-\frac{V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}$$

How do I use this to find $\omega$?

 

[SGT]

If you want Vo to be zero, the impedance of the series connection of the capacitor with the inductor must be zero.
You got the impedance of the capacitor wrong.
$$Z_C = \frac{-j}{\omega C}$$
$$Z_L = j \omega L$$

 

[VinnyCee]

http://img99.imageshack.us/img99/4333/problem934part2xc6.jpg

$$Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega$$

$$V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}$$

We want $V_{OUT}$ to be zero:

$$0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,20j\omega\,+\,2\Omega}\,V_{IN}$$

$$0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}$$

$$-\frac{1}{5j\omega}\,=\,20j\omega$$

$$\omega\,=\,\frac{1}{10}$$

Is that right?

 

[SGT]

http://img99.imageshack.us/img99/4333/problem934part2xc6.jpg

$$Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega$$

$$V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}$$

We want $V_{OUT}$ to be zero:

$$0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,12j\omega\,+\,2\Omega}\,V_{IN}$$

$$0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}$$

$$-\frac{1}{5j\omega}\,=\,20j\omega$$

$$\omega\,=\,\frac{1}{10}$$

Is that right?

Yes, the impedance is zero at the resonant frequency $$\omega_0 = \frac{1}{(LC)^\frac{1}{2}} = \frac{1}{10}$$