✎ Pendulum [The Physics Travel Guide]

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<WRAP lag>$ \quad L = \frac{1}{2} \color{blue}{m} \color{olive}{l}^2\dot{\color{red}{\phi}}^2 - \color{blue}{m}\color{magenta}{g}\color{olive}{l} (1-cos \color{red}{\phi})$</WRAP>

====== Pendulum ======

A pendulum is right after a harmonic oscillator the simplest physical system we can study. In fact, if the pendulum only swings a little it is a [[models:harmonic_oscillator|harmonic oscillator]]. The difference between the harmonic oscillator and the pendulum only become important for large swings.

A pendulum consists of a freely hanging massive bob at the end of a rod. When we move the bob a little to one side it starts swinging.

We usually describe it by measuring $\color{red}{\text{how far the bob has moved from its original position}}$ where it just hangs freely. How the pendulum swings depends crucially on the $\color{olive}{\text{length of the rod}}$ and the $\color{magenta}{\text{strength of the gravitational field}}$. A pendulum on the moon swings differently than a pendulum on earth.

An important observation is that the swinging of the pendulum does not depend on the $\color{blue}{\text{mass of the bob }}$.

<tabbox Concrete> 
{{ :models:pendulum.png?nolink&300|}}
[[models:harmonic_oscillator|]]
A normal pendulum hangs freely in a uniform gravitational field of strength $g$ on a rod with length $l$. The excitation above the ground state is measured by the angle $\phi$. At the end of the pendulum, we have a bob of mass $m$. This is shown in the picture on the right-hand side.

The only force that acts on the pendulum is the gravitational force $F= mg \sin\phi $. Therefore, using Newton's second law the correct equation of motion for the pendulum is

$$m\frac{d^2 x}{dt^2}= - mg \sin \phi .$$

Now we need to get rid of the arc length on the left-hand side using $x = l \phi$. This yields

$$m\frac{d^2 x}{dt^2}= - m\frac{g}{l} \sin \phi .$$

It is important to note that we can remove mass $m$ from the equation because it appears on both sides:

$$\frac{d^2 x}{dt^2}= - \frac{g}{l} \sin \phi .$$

The swinging of the pendulum, therefore, does not depend on the mass of the bob.

----
**Small Angle Approximation**

The equation of motion of the pendulum is a non-linear equation which is hard to solve. However we can simplify it for the case when only small swings are happening. Then we can use the small angle approximation $\sin({\theta)} \approx \theta$.

This yields the equation \[\frac{d^2\theta}{dt^2} = -\frac{g}{L}\theta\] which can be solved to give the familiar equation for the harmonic oscillator \[\theta = A\sin({\beta t + \phi}), \beta = \sqrt{\frac{g}{l}}\]
where $A$ is the amplitude and $\phi$ is the constant phase offset.

----

**The Pendulum in the Lagrangian Formalism**

The __kinetic energy__ of the pendulum is

$$T=\frac{1}{2}ml^2(\frac{d\theta}{dt})^2$$

and the __potential energy__ in the gravitational field of the earth

$$ U=mgl(1-cos\theta).$$

The [[formalisms:lagrangian_formalism|Lagrangian]] of the pendulum is therefore

$$ L = T-V= \frac{1}{2} ml^2\dot{\phi}^2 - mgl (1-cos \phi), $$
where $\dot{\phi}\equiv d\phi /dt $ denotes the time derivative.

Using the Euler-Lagrange equation we can derive the corresponding __equation of motion__

$$\frac{d^2\theta}{dt^2}+\frac{g}{l}sin\theta=0 .$$

----

* For a nice complete discussion, see The Pendulum: A Case Study in Physics by Gregory L. Baker and James A. Blackburn
<tabbox Abstract>

* A thorough discussion can be found in [[https://aapt.scitation.org/doi/abs/10.1119/1.12332|Quantum pendulum]] by R. Aldrovandi and P. Leal Ferreira

<tabbox Why is it interesting?> 
The pendulum is one of the most important basic physics systems.

Understanding the pendulum is crucial to understand topics as advanced as, for example, [[advanced_notions:quantum_field_theory:solitons|Solitons]] or the [[advanced_notions:quantum_field_theory:qcd_vacuum|QCD vacuum]].

In addition, it is ideal to understand the notion [[basic_tools:dimensional_analysis|dimensional analysis]]. 
  
<tabbox History>

* Time for Science Education - How Teaching the History and Philosophy of Pendulum Motion can Contribute to Science Literacy by Michael R. Matthews

</tabbox>

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