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--- 183_notes:curving_motion [2014/10/06 18:06] – caballero
+++ 183_notes:curving_motion [2015/02/18 23:45] – [Relationship to the tangential and centripetal accelerations] obsniukm
@@ Line 73: / Line 73: @@
 ==== Relationship to the tangential and centripetal accelerations ====
-In your previous studies, you might come acres the [[http://en.wikipedia.org/wiki/Acceleration#Tangential_and_centripetal_acceleration|tangential acceleration ($\vec{a}_{t}$) and the centripetal acceleration ($\vec{a}_{c}$)]]. These are directly connected to the definitions of the parallel and perpendicular components of the net force. You can write the net force as the sum of these parallel and perpendicular components, which arise from the tangential and centripetal accelerations.
+In your previous studies, you might have come across the [[http://en.wikipedia.org/wiki/Acceleration#Tangential_and_centripetal_acceleration|tangential acceleration ($\vec{a}_{t}$) and the centripetal acceleration ($\vec{a}_{c}$)]]. These are directly connected to the definitions of the parallel and perpendicular components of the net force. You can write the net force as the sum of these parallel and perpendicular components, which arise from the tangential and centripetal accelerations.
 $$\vec{F}_{net} = \vec{F}_{\parallel} + \vec{F}_{\perp}$$
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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.
-$$F_{\parallel}  = m{a}_{t} = \dfrac{d|\vec{p}|}{dt} = \dfrac{d|m\vec{v}|}{dt} = m\dfrac{d|\vec{v}|}{dt} \longrightarrow {a}_{t} = \dfrac{d|\vec{v}|}{dt}$$
+$$F_{\parallel}  = m{a}_{t} = \dfrac{d|\vec{p}|}{dt} = \dfrac{d|m\vec{v}|}{dt} = m\dfrac{d|\vec{v}|}{dt} \qquad\longrightarrow\qquad {a}_{t} = \dfrac{d|\vec{v}|}{dt}$$
 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.
-$$F_{\perp}  = m{a}_{c} = \dfrac{|\vec{p}||\vec{v}|}{R} = \dfrac{mv^2}{R} = m\dfrac{v^2}{R} \longrightarrow {a}_{c} = \dfrac{v^2}{R}$$
+$$F_{\perp}  = m{a}_{c} = \dfrac{|\vec{p}||\vec{v}|}{R} = \dfrac{mv^2}{R} = m\dfrac{v^2}{R}\qquad\longrightarrow\qquad {a}_{c} = \dfrac{v^2}{R}$$
 The centripetal acceleration tells you how the direction of the object's motion changes, just as the perpendicular component of the net force is responsible for this directional change.