User Tools

Site Tools


models:basic_models:pendulum

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
models:basic_models:pendulum [2018/04/09 08:48]
lushikatome [Concrete]
models:basic_models:pendulum [2020/04/02 15:37] (current)
75.97.173.144
Line 1: Line 1:
-<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})$</​WRAP>​+<WRAP lag>$ \quad L = \frac{1}{2} \color{blue}{m} \color{olive}{l}^2\dot{\color{firebrick}{\phi}}^2 - \color{blue}{m}\color{magenta}{g}\color{olive}{l} (1-cos \color{firebrick}{\phi})$</​WRAP>​
  
 ====== Pendulum ====== ====== Pendulum ======
Line 8: Line 8:
 {{ :​models:​pendulumsimple.png?​nolink&​200|}} {{ :​models:​pendulumsimple.png?​nolink&​200|}}
  
-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. ​ 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.+We usually describe it by measuring ​<color firebrick>​how far the bob has moved from its original position</​color> ​where it just hangs freely. How the pendulum swings depends crucially on the <color olive>length of the rod</​color> ​and the <color magenta>strength of the gravitational field</​color>​. 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>​mass of the bob</​color>​. 
 + 
 +A pendulum is right after a [[models:​basic_models:​harmonic_oscillator|harmonic oscillator]] the simplest physical system we can study. In fact, if the pendulum only swings a little it is a harmonic oscillator. The difference between the harmonic oscillator and the pendulum only become important for large swings.  
 + 
 +----
  
-An important observation is that the swinging of the pendulum ​does not depend on the $\color{blue}{\text{mass of the bob }}$.+  * see also the nice interactive demonstration [[https://​www.myphysicslab.com/​pendulum/​pendulum-en.html|here]]
  
   ​   ​
 <tabbox Concrete> ​ <tabbox Concrete> ​
 {{ :​models:​pendulum.png?​nolink&​300|}} {{ :​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.  ​ 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.  ​
  
Line 59: Line 65:
 The [[formalisms:​lagrangian_formalism|Lagrangian]] of the pendulum is therefore The [[formalisms:​lagrangian_formalism|Lagrangian]] of the pendulum is therefore
  
-$$ L = T-V= \frac{1}{2} ml^2\dot{\phi}^2 - mgl (1-cos \phi), $$ +$$ L = T-U= \frac{1}{2} ml^2\dot{\theta}^2 - mgl (1-cos \theta), $$ 
-where $\dot{\phi}\equiv d\phi /dt $ denotes the time derivative. ​+where $\dot{\theta}\equiv d\theta /dt $ denotes the time derivative. ​
  
 Using the Euler-Lagrange equation we can derive the corresponding __equation of motion__ Using the Euler-Lagrange equation we can derive the corresponding __equation of motion__
Line 71: Line 77:
   * For a nice complete discussion, see The Pendulum: A Case Study in Physics by Gregory L. Baker and James A. Blackburn   * For a nice complete discussion, see The Pendulum: A Case Study in Physics by Gregory L. Baker and James A. Blackburn
 <tabbox Abstract> ​ <tabbox Abstract> ​
 +The [[basic_tools:​phase_space|phase space]] of a pendulum
 +
 +{{ :​basic_tools:​phasespacependulum2.png?​nolink&​600 |}}
 +
 +
 +----
 +
 +
  
   * 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   * 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
models/basic_models/pendulum.1523256535.txt.gz · Last modified: 2018/04/09 06:48 (external edit)