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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 18:02] – [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: |
- | $$\dfrac{d \hat{p}}{d t} = \dfrac{v_{avg}}{R} \hat{n}$$ | + | $$\dfrac{d \hat{p}}{d t} = \dfrac{|\vec{v}|}{R} \hat{n}$$ |
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+ | [{{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" | ||
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+ | $$\vec{F}_{net, | ||
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+ | ==== Relationship to the tangential and centripetal accelerations ==== | ||
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+ | In your previous studies, you might come acres the [[http:// | ||
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+ | $$\vec{F}_{net} = \vec{F}_{\parallel} + \vec{F}_{\perp}$$ | ||
+ | $$\vec{F}_{\parallel} | ||
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+ | The direction of each of these accelerations is the same as their corresponding forces. The tangential acceleration is tangent to the path, and this points in the $\hat{p}$ direction. The centripetal acceleration is perpendicular to the path and points in the $\hat{n}$ direction. You can use the magnitudes of each force component to determine formulae for the accelerations. | ||
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+ | $$F_{\parallel} | ||
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+ | The tangential acceleration tells you how the speed of the object changes, just as the parallel component of the net force is responsible for this speeding up and slowing down. | ||
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+ | $$F_{\perp} | ||
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+ | The centripetal acceleration tells you how the direction of the object' | ||