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

• Andrei0408
In summary: So, yeah, I guess LC circuits could technically move! Good luck with that problem!In summary, the equations for simple harmonic motion show that the natural frequency, or the frequency at which a system oscillates spontaneously, is given by:
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!

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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!

Andrei0408 said:
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/

berkeman said:
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.

berkeman
(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!

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?

etotheipi said:
##\dots~##to derive the EoM for the LC circuit?
Do LC circuits move?

etotheipi
kuruman said:
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)##.

## 1. How does a pendulum's motion compare to that of an LC circuit?

The motion of a pendulum and an LC circuit can be compared by looking at their respective equations of motion. While a pendulum's motion is described by a simple harmonic motion equation, an LC circuit's motion is described by a damped harmonic motion equation. This means that while both systems exhibit periodic motion, the pendulum's amplitude will gradually decrease due to friction, while the LC circuit's amplitude will decrease due to energy dissipation through the resistor.

## 2. What are the key components of an LC circuit?

An LC circuit consists of an inductor (L) and a capacitor (C) connected in series. The inductor stores energy in the form of a magnetic field, while the capacitor stores energy in the form of an electric field. These two components work together to create the oscillatory motion in the circuit.

## 3. How is energy dissipated in an LC circuit?

In an LC circuit, energy is dissipated through the resistor (R) in the circuit. This resistor converts the electrical energy into heat energy, causing the amplitude of the oscillations to decrease over time.

## 4. What factors affect the period of oscillation in an LC circuit?

The period of oscillation in an LC circuit is affected by the values of the inductance (L) and capacitance (C) in the circuit. A larger inductance or capacitance will result in a longer period of oscillation, while a smaller inductance or capacitance will result in a shorter period of oscillation.

## 5. How can the motion of an LC circuit be visualized?

The motion of an LC circuit can be visualized by using an oscilloscope. This device displays the voltage across the capacitor over time, showing the oscillatory motion of the circuit. Alternatively, a physical model or simulation can also be used to visualize the motion of an LC circuit.

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