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}