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183_notes:ap_derivation [2014/11/20 17:54] – 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 ==== | ||
- | $$F_{net} = \dfrac{d\vec{p}}{dt}$$ | + | Consider a single particle (mass, |
+ | |||
+ | $$\vec{F}_{net} = \dfrac{d\vec{p}}{dt}$$ | ||
+ | |||
+ | Now, if we consider the cross product of the momentum principle with some defined lever arm (e.g., the origin of coordinates), | ||
$$\vec{r} \times \vec{F}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$ | $$\vec{r} \times \vec{F}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$ | ||
+ | |||
+ | This cross product of the lever arm and the net force is the net torque about that chosen location, | ||
$$\vec{\tau}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$ | $$\vec{\tau}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$ | ||
+ | |||
+ | The right hand-side of the equation can be re-written using the [[http:// | ||
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \dfrac{d\vec{r}}{dt} \times \vec{p}$$ | $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \dfrac{d\vec{r}}{dt} \times \vec{p}$$ | ||
+ | |||
+ | The term on the far right is the cross product of the particle' | ||
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \vec{v} \times \vec{p}$$ | $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \vec{v} \times \vec{p}$$ | ||
+ | |||
+ | which for an object that doesn' | ||
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - m \underbrace{\vec{v} \times \vec{v}}_{=0}$$ | $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - m \underbrace{\vec{v} \times \vec{v}}_{=0}$$ | ||
+ | |||
+ | And thus, we have the angular momentum principle in its derivative form, | ||
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right)$$ | $$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right)$$ | ||
$$\vec{\tau}_{net} = \dfrac{d\vec{L}}{dt}$$ | $$\vec{\tau}_{net} = \dfrac{d\vec{L}}{dt}$$ | ||
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