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| 183_notes:ap_derivation [2014/11/20 18:06] – caballero | 183_notes:ap_derivation [2014/11/20 18:06] (current) – caballero | ||
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| ===== Derivation of the Angular Momentum Principle ==== | ===== Derivation of the Angular Momentum Principle ==== | ||
| - | Consider a single particle (mass, $m$) that is moving with a momentum $p$. This particle experiences a net force $F_{net}$, which will change the particle' | + | Consider a single particle (mass, $m$) that is moving with a momentum $\vec{p}$. This particle experiences a net force $\vec{F}_{net}$, which will change the particle' |
| - | $$F_{net} = \dfrac{d\vec{p}}{dt}$$ | + | $$\vec{F}_{net} = \dfrac{d\vec{p}}{dt}$$ |
| Now, if we consider the cross product of the momentum principle with some defined lever arm (e.g., the origin of coordinates), | Now, if we consider the cross product of the momentum principle with some defined lever arm (e.g., the origin of coordinates), | ||
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| $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - m \underbrace{\vec{v} \times \vec{v}}_{=0}$$ | $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - m \underbrace{\vec{v} \times \vec{v}}_{=0}$$ | ||
| - | And thus, we have the angular momentum principle in it' | + | And thus, we have the angular momentum principle in its derivative form, |
| $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right)$$ | $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right)$$ | ||