Equation Demonstration -- Comparing a pendulum's motion to an LC circuit

[Andrei0408]

Homework Statement

I need the demonstration of the attached equations.

Relevant Equations

I've attached the 2 equations.

I've just learned about simple harmonic motion and I've been given the following examples: The physical pendulum (for small oscillations sin(theta)~theta), with the formula (1st pic), and the LC circuit, with the formula (2nd pic). If possible, I need the demonstration for these 2 formulas. Thank you!

 

[etotheipi]

For the first one, set $\tau_z = I_z \ddot{\theta}_z$. For the second one, write out Kirchoff's law for the circuit. Unfortunately, the laws of Physics forums prevent me from helping any further until you've given it a shot!

 

[berkeman]

Homework Statement:: I need the demonstration of the attached equations.
Relevant Equations:: I've attached the 2 equations.

I've just learned about simple harmonic motion and I've been given the following examples: The physical pendulum (for small oscillations sin(theta)~theta), with the formula (1st pic), and the LC circuit, with the formula (2nd pic). If possible, I need the demonstration for these 2 formulas. Thank you!

You know that you are required to show your efforts to work the problem before we can offer tutorial help. See what you can find as references for those two situations please.

Also, you never replied in your other thread about the car and the banked turn. What did you end up finding on that problem?

https://www.physicsforums.com/threa...ed-turn-in-a-road-for-a-maximum-speed.994630/

 

[Andrei0408]

You know that you are required to show your efforts to work the problem before we can offer tutorial help. See what you can find as references for those two situations please.

Also, you never replied in your other thread about the car and the banked turn. What did you end up finding on that problem?

https://www.physicsforums.com/threa...ed-turn-in-a-road-for-a-maximum-speed.994630/

Sorry I didn't reply faster, I had a lot of uni work. Yes I did manage to solve it, by adding half of the angle of the friction cone to the equation, then I used trigonometric functions to find out theta.

 

[callieferg]

(1) The general form for the natural frequency $ω_0$ (or at least the way I learned it) is: $ω_0=\sqrt\frac{k}{m}$
(2) Look at the differential equations for Newton's second law and Kirchoff's second rule for circuits. If you can find the differences in the coefficients for each of these equations, it might help in understanding how the two formulas you provided are related to each other. Good luck!

 

[etotheipi]

Sometimes you can derive these things nicely by considering energy, for instance for the pendulum$$\begin{align*}E &= \frac{1}{2}mr^2 \dot{\theta}^2 - mgr\cos{\theta} \\ \frac{dE}{dt} &= mr^2 \ddot{\theta}\dot{\theta} + mgr \dot{\theta} \sin{\theta} \approx mr^2 \dot{\theta} \ddot{\theta} + mgr \theta \dot{\theta} \\ \\ \frac{dE}{dt} &= 0 \implies \ddot{\theta} \approx -\frac{g}{r}\theta \end{align*}$$Can you use the same approach, with $E = \frac{1}{2}CV^2 + \frac{1}{2}Li^2$, to derive the EoM for the LC circuit?

 

[kuruman]

$\dots~$to derive the EoM for the LC circuit?

Do LC circuits move?

 

[etotheipi]

Do LC circuits move?

Ahaha, you never know! My differential equations prof. uses "equation of motion" to refer to any solution that is a function of time, e.g. $i(t)$.