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183_notes:curving_motion [2014/09/24 17:18] – [A change in direction] caballero | 183_notes:curving_motion [2014/09/24 17:51] – [Relationship to the tangential and centripetal accelerations] caballero | ||
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As [[183_notes: | As [[183_notes: | ||
- | $$\dfrac{d \hat{p}}{d t} = \dfrac{v_{avg}}{R\theta} \langle -\theta, 0\rangle = \dfrac{v_{avg}}{R} \langle -1, 0\rangle$$ | + | $$\dfrac{d \hat{p}}{d t} = \dfrac{|\vec{v}|}{R\theta} \langle -\theta, 0\rangle = \dfrac{|\vec{v}|}{R} \langle -1, 0\rangle$$ |
- | where you are using the approximation for small $\theta$ in both the cosine ($\cos \theta \approx 1$) and sine (#\sin \theta \approx \theta$). This vector points in the $-x$-dreiction, which is toward the turn and perpendicular to the direction of the momentum vector. This result generalizes to: | + | where you are using the approximation for small $\theta$ in both the cosine ($\cos \theta \approx 1$) and sine ($\sin \theta \approx \theta$). Here, the average velocity is replaced by the instantaneous because you are looking at infinitesimally short time interval. This vector points in the $-x$-direction, which is toward the turn and perpendicular to the direction of the momentum vector. This result generalizes to: |
- | $$\dfrac{d \hat{p}}{d t} = \dfrac{v_{avg}}{R} \hat{n}$$ | + | $$\dfrac{d \hat{p}}{d t} = \dfrac{|\vec{v}|}{R} \hat{n}$$ |
+ | |||
+ | [{{183_notes: | ||
where the unit vector, $\hat{n}$, always points inward towards the turn. For more general trajectories, | where the unit vector, $\hat{n}$, always points inward towards the turn. For more general trajectories, | ||
+ | The change in the direction of the momentum is the result of the component of the net force that is perpendicular to the direction of motion (momentum). This component is referred to as "F net perpendicular" | ||
+ | |||
+ | $$\vec{F}_{net, | ||
+ | |||
+ | ==== Relationship to the tangential and centripetal accelerations ==== | ||
+ | In your previous studies, you might come acres the [[http:// | ||
+ | $$\vec{F}_{net} = \vec{F}_{\parallel} + \vec{F}_{\perp}$$ | ||
+ | $$\vec{F}_{\parallel} |