What is the frequency of the source in the given RLC circuit?

[Misaki]

Homework Statement

In the AC circuit shown, the potential difference across the capacitor and the resistor, V(BD), is 24 V(rms). Similarly, the potential difference across the inductor and the resistor, V(AC) is 24 V(rms). What is the frequency of the source?

http://img52.imageshack.us/img52/8039/lolkve.png

Homework Equations

V = sqrt(V(R)^2 + V(L)^2)
V = sqrt((I(rms)*R)^2 + (I(rms)*X(L))^2)

V = sqrt(V(R)^2 + V(C)^2)
V = sqrt((I(rms)*R)^2 + (I(rms)*X(C))^2)

X(L) = 2pi*f*L
X(C) = 1/(2pi*f*C)

The Attempt at a Solution

So, immediately I saw that I could use the two voltage equations above to solve for either I(rms) or f, since both are constants. I decided to solve for f, as it is what the question is asking for. I arbitrarily took the inductance equation and rearranged for I(rms).

http://img201.imageshack.us/img201/7150/codecogseqn1s.gif
http://img259.imageshack.us/img259/1027/codecogseqn2w.gif
http://img138.imageshack.us/img138/8778/codecogseqn3.gif
http://img3.imageshack.us/img3/3735/codecogseqn9.gif

Plugging that into the other equation:

http://img705.imageshack.us/img705/7231/codecogseqn5x.gif

Simplified into:

http://img252.imageshack.us/img252/7870/codecogseqn7.gif

http://img849.imageshack.us/img849/9776/codecogseqn8.gif

Essentially, I ended up with something like:

http://img710.imageshack.us/img710/2517/codecogseqn10.gif

It's at this point that I basically gave up. I plugged the equation into wolframalpha, got around 89 Hz, and decided that that was a reasonable number.

Now, my question is, did I approach the problem incorrectly? Or is this the right way to do it, and the numbers are ACTUALLY this complicated? Or did I just make an algebraic error somewhere? Any help would be appreciated.

There are 3 more questions, but I need the frequency in order to solve them. Once I have it, the other 3 questions can be easily solved.

 

[gneill]

Homework Statement

In the AC circuit shown, the potential difference across the capacitor and the resistor, V(BD), is 24 V(rms). Similarly, the potential difference across the inductor and the resistor, V(AC) is 24 V(rms). What is the frequency of the source?

http://img52.imageshack.us/img52/8039/lolkve.png

Homework Equations

V = sqrt(V(R)^2 + V(L)^2)
V = sqrt((I(rms)*R)^2 + (I(rms)*X(L))^2)

V = sqrt(V(R)^2 + V(C)^2)
V = sqrt((I(rms)*R)^2 + (I(rms)*X(C))^2)

X(L) = 2pi*f*L
X(C) = 1/(2pi*f*C)

The Attempt at a Solution

So, immediately I saw that I could use the two voltage equations above to solve for either I(rms) or f, since both are constants. I decided to solve for f, as it is what the question is asking for. I arbitrarily took the inductance equation and rearranged for I(rms).

http://img201.imageshack.us/img201/7150/codecogseqn1s.gif
http://img259.imageshack.us/img259/1027/codecogseqn2w.gif
http://img138.imageshack.us/img138/8778/codecogseqn3.gif
http://img3.imageshack.us/img3/3735/codecogseqn9.gif

Plugging that into the other equation:

http://img705.imageshack.us/img705/7231/codecogseqn5x.gif

Simplified into:

http://img252.imageshack.us/img252/7870/codecogseqn7.gif

http://img849.imageshack.us/img849/9776/codecogseqn8.gif

Essentially, I ended up with something like:

http://img710.imageshack.us/img710/2517/codecogseqn10.gif

It's at this point that I basically gave up. I plugged the equation into wolframalpha, got around 89 Hz, and decided that that was a reasonable number.

Now, my question is, did I approach the problem incorrectly? Or is this the right way to do it, and the numbers are ACTUALLY this complicated? Or did I just make an algebraic error somewhere? Any help would be appreciated.

There are 3 more questions, but I need the frequency in order to solve them. Once I have it, the other 3 questions can be easily solved.

Hi Misaki. Welcome to Physics Forums.

Wow, you did a lot of "heavy lifting" for what could be a straight forward problem

If this problem had involved pure resistances (suppose there's an RC and and RL in place of the capacitor and inductor), and given the described potential differences, what would you have concluded about the values of RC and RL?

 

[Misaki]

Hi Misaki. Welcome to Physics Forums.

Wow, you did a lot of "heavy lifting" for what could be a straight forward problem

If this problem had involved pure resistances (suppose there's an RC and and RL in place of the capacitor and inductor), and given the described potential differences, what would you have concluded about the values of RC and RL?

Well, if they were pure resistances, and you're given the potential difference across them, you could just calculate an equivalent resistance and use that to solve for current, right? But since they aren't just pure resistances, and you're missing f (which is needed for ω), you have two missing variables instead of 1. Am I missing something here?

 

[gneill]

if they were pure resistances, what would you conclude about the values of RC and RL?

 

[Misaki]

if they were pure resistances, what would you conclude about the values of RC and RL?

I thought about that for around 10 minutes, but I seriously don't know the answer.

 

[gneill]

I thought about that for around 10 minutes, but I seriously don't know the answer.

Umm. If A+B = B+C, what's the relationship between A and C?

 

[Misaki]

Umm. If A+B = B+C, what's the relationship between A and C?

OH.

Wow, I can't believe I didn't realize that.
So essentially, it's at resonance, right?

 

[gneill]

OH.

Wow, I can't believe I didn't realize that.
So essentially, it's at resonance, right?

Huzzah! Yes!

 

[Misaki]

Huzzah! Yes!

Thanks, haha. I feel really dumb now. At least I got the right answer with my approach. xD