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models:basic_models:pendulum [2018/05/15 06:50]
jakobadmin
models:basic_models:pendulum [2020/04/02 15:37] (current)
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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{firebrick}{\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 ======
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 {{ :​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:​basic_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. ​
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 An important observation is that the swinging of the pendulum does not depend on the <color blue>​mass of the bob</​color>​. 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. ​
  
 ---- ----
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 <tabbox Concrete> ​ <tabbox Concrete> ​
 {{ :​models:​pendulum.png?​nolink&​300|}} {{ :​models:​pendulum.png?​nolink&​300|}}
-[[models:​basic_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.  ​
  
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 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__
models/basic_models/pendulum.1526359853.txt.gz · Last modified: 2018/05/15 04:50 (external edit)