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models:basic_models:pendulum [2018/05/12 13:29] jakobadmin [Abstract] |
models:basic_models:pendulum [2020/04/02 15:37] (current) 75.97.173.144 |
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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})$</WRAP> | + | <WRAP lag>$ \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. | ||
- | 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 $\color{blue}{\text{mass of the bob }}$. | + | 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__ | ||
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* 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 phase space of a pendulum | + | The [[basic_tools:phase_space|phase space]] of a pendulum |
{{ :basic_tools:phasespacependulum2.png?nolink&600 |}} | {{ :basic_tools:phasespacependulum2.png?nolink&600 |}} |