LRC Series Circuit Critical Damping

[rugerts]

Homework Statement

Homework Equations

The Attempt at a Solution

My attempted solution is above and here https://imgur.com/8RmDMf8/
I'm confused as to the answers in the book being i and iii (I just don't see how i is included). If critical damping occurs at the value above, and if you go above you've got over damping and below underdamping, I don't see how my math isn't working out. I must be missing something. Any advice? Thanks.

 

[gneill]

You're misapplying the conditional test.

Overdamping occurs when $R^2 > \frac{4L}{C}$, which it certainly is in the first case.

Underdamping occurs when $R^2 < \frac{4L}{C}$, which it certainly is in the second case.

 

[rugerts]

You're misapplying the conditional test.

Overdamping occurs when $R^2 > \frac{4L}{C}$, which it certainly is in the first case.

Underdamping occurs when $R^2 < \frac{4L}{C}$, which it certainly is in the second case.

What am I missing?
My R^2 value for case i was 2 which is less than 4. Underdamping
My R^2 value for case ii was 8 which is greater than 4. Overdamping

 

[gneill]

What am I missing?
My R^2 value for case i was 2 which is less than 4. Underdamping
My R^2 value for case ii was 8 which is greater than 4. Overdamping

R^2 is fixed at 4. You compare the value of R² to 4L/C.

i) Since R² > 4L/C, the circuit is overdamped.
ii) Since R² < 4L/C, the circuit is underdamped.

 

[rugerts]

R^2 is fixed at 4. You compare the value of R² to 4L/C.

i) Since R² > 4L/C, the circuit is overdamped.
ii) Since R² < 4L/C, the circuit is underdamped.

Oh, I see. Thank you.

 

[gneill]

You're very welcome. Happy to help.