basic_notions:energy
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basic_notions:energy [2017/12/04 08:01] 127.0.0.1 external edit |
basic_notions:energy [2018/04/12 16:50] bogumilvidovic [Concrete] |
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| ====== Energy ====== | ====== Energy ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | <blockquote>Energy is the most important concept in physics<cite>[[https://gravityandlevity.wordpress.com/2010/01/30/the-universe-is-a-giant-energy-minimization-machine/|Brian Skinner]]</cite></blockquote> | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| * https://gravityandlevity.wordpress.com/2009/04/13/force-and-energy-which-is-more-real/ | * https://gravityandlevity.wordpress.com/2009/04/13/force-and-energy-which-is-more-real/ | ||
| Line 12: | Line 10: | ||
| * https://gravityandlevity.wordpress.com/2009/05/16/the-equivalence-of-mass-and-energy-the-center-of-energy/ | * https://gravityandlevity.wordpress.com/2009/05/16/the-equivalence-of-mass-and-energy-the-center-of-energy/ | ||
| | | ||
| - | <tabbox Student> | + | <tabbox Concrete> |
| - | <note tip> | + | Energy is the conserved quantity that we derive using Noether's theorem if our system is symmetric under temporal translations. |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | |
| - | </note> | + | |
| - | + | ||
| - | <tabbox Researcher> | + | |
| - | <note tip> | + | In addition, energy is responsible for temporal translations. We say energy generates temporal translations. |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
| - | | ||
| - | <tabbox Examples> | ||
| - | --> Example1# | + | The total energy is defined as |
| + | \begin{equation} | ||
| + | E(t) \equiv K(t) + V(q(t)), | ||
| + | \end{equation} | ||
| - | + | where $K$ denotes the kinetic energy and $V$ the potential energy. | |
| - | <-- | + | |
| - | --> Example2:# | + | -->Proof the the total energy is conserved# |
| + | |||
| + | For a system with a conservative force the relationship between force and potential energy is given by $ | ||
| + | \grad V \equiv - F$. | ||
| + | |||
| + | In addition, [[equations:newtons_second_law|Newton's second law]] $F = ma$ implies | ||
| + | \[ | ||
| + | \begin{split} | ||
| + | \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) + | ||
| + | \grad V(q(t))\cdot v(t) \\ | ||
| + | &= 0, \qquad\text{(because $F=-\grad V$)}. | ||
| + | \end{split} | ||
| + | \] | ||
| - | |||
| <-- | <-- | ||
| + | ---- | ||
| + | |||
| + | |||
| + | **Kinetic Energy** | ||
| + | |||
| + | Kinetic energy is defined as | ||
| + | \begin{equation} | ||
| + | K(t) \equiv \frac{1}{2}m\,v(t)\cdot v(t). | ||
| + | \end{equation} | ||
| + | This quantity is useful because | ||
| + | \[ | ||
| + | \begin{split} | ||
| + | \frac{d}{dt}K(t) &= m\,v(t)\cdot a(t) \\ | ||
| + | &= F(q(t))\cdot v(t). | ||
| + | \end{split} | ||
| + | \] | ||
| + | We can see here that the kinetic energy goes up whenever we push an object in the direction | ||
| + | of its velocity. Moreover, it goes down whenever we push it in the opposite | ||
| + | direction. | ||
| + | |||
| + | In addition, we have | ||
| + | \[ | ||
| + | \begin{split} | ||
| + | K(t_1)-K(t_0) &= \int_{t_0}^{t_1} F(q(t))\cdot v(t)\,dt \\ | ||
| + | &= \int_{t_0}^{t_1} F(q(t))\cdot \dot{q}(t)\, dt. | ||
| + | \end{split} | ||
| + | \] | ||
| + | This tells us that the change of kinetic energy is equal to the __work__ done by the | ||
| + | force. The work is defined as the integral of $F$ along the trajectory. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | **Potential Energy** | ||
| + | |||
| + | $ | ||
| + | \grad V \equiv - F,$ | ||
| + | |||
| + | where $F$ denotes the force. | ||
| + | <tabbox Abstract> | ||
| + | |||
| + | <note tip> | ||
| + | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| + | </note> | ||
| - | <tabbox FAQ> | ||
| | | ||
| - | <tabbox History> | + | <tabbox Why is it interesting?> |
| + | |||
| + | <blockquote>Energy is the most important concept in physics<cite>[[https://gravityandlevity.wordpress.com/2010/01/30/the-universe-is-a-giant-energy-minimization-machine/|Brian Skinner]]</cite></blockquote> | ||
| </tabbox> | </tabbox> | ||
basic_notions/energy.txt · Last modified: 2018/04/12 16:51 by bogumilvidovic
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