Critically damped current in this network

[EEZeroo]

Homework Statement

I'm supposed to find whether the current in the following network is

a) under-damped
b) critically damped
c) undamped
c) over-damped

Homework Equations

According to Kirchoff's laws

The Attempt at a Solution

The answer given by my professor for this question is supposed to be "b) critically damped" so I just converted the given values into SI values
R = 1kilo-ohm = 1000 ohm
L = 10mH = 0,01H
C = 40nF = 4 * 10^(-8)

and then I tried relating them to

But I do not find two real roots that are equal for it to be critically damped.

What did I do wrong?

 

[antny]

Critical damping will occur when [itex]\alpha[/itex][itex]^{2}[/itex]-4[itex]\omega^{2}_{o}[/itex]=0. Another way to write it is: R=sqrt(4*L/C). If you plug in your values for R, L and C, you'll find that it is indeed critically damped.

 

[EEZeroo]

Thank you. I see that my silly mistake was that I substracted the 10 instead of leaving it homogenous.

Though, when exactly is a circuit undamped? I know when it's critically, overdamped or underdamped but when would it be undamped?

 

[antny]

If there's a resistor, there will always be energy lost, hence the damping.
An LC circuit (no resistor) will oscillate without damping.

 

[DeeNos]

As a scientist, it is important to critically analyze your calculations and assumptions when faced with a discrepancy in your results. In this case, it seems that you may have made a mistake in your conversion of the given values into SI units. Let's take a closer look at your calculations to see if we can identify the error.

Firstly, your conversion of 1 kilo-ohm to 1000 ohm is correct. However, your conversion of 10mH to 0.01H is incorrect. The correct conversion would be 10mH = 0.01H = 10 * 10^(-3)H. Similarly, your conversion of 40nF to 4 * 10^(-8)F is also incorrect. The correct conversion would be 40nF = 4 * 10^(-8)F = 40 * 10^(-9)F.

Now, let's plug in the correct values into the equation for the critically damped condition:

Δ = R^2 - 4L/C

= (1000)^2 - 4*(0.01)*(40*10^(-9))

= 1000000 - 4*(0.0004)

= 1000000 - 0.0016

= 999999.9984

As you can see, the value of Δ is very close to zero, but not exactly zero. This could be due to rounding errors in your calculations or the given values themselves. However, it is important to note that for a system to be critically damped, the value of Δ must be exactly zero. Therefore, it is possible that your professor may have made a mistake in their answer or the given values may not actually lead to a critically damped system.

As a scientist, it is important to always double-check your calculations and assumptions and to not blindly accept given answers. In this case, it may be beneficial to discuss your findings with your professor and see if they can provide further clarification. Additionally, you could also try to solve the problem using different methods or formulas to see if you arrive at the same result.