Differences

This shows you the differences between two versions of the page.

--- 183_notes:curving_motion [2014/09/29 19:32] – pwirving
+++ 183_notes:curving_motion [2021/03/04 12:56] (current) – [Modeling Curved Motion] stumptyl
@@ Line 1: / Line 1: @@
+Section 5.5, 5.6 and 5.7 in Matter and Interactions (4th edition)
 ===== Modeling Curved Motion =====
-The motion of objects is not limited to [[183_notes:displacement_and_velocity|straight line motion]]. As you read earlier, [[183_notes:momentum_principle|forces can change the momentum of objects]] (including the direction of that momentum). These interactions can produce [[183_notes:localg|projectile motion]], [[183_notes:ucm|circular motion]], [[183_notes:springmotion|oscillations]], or more generalized trajectories. In these notes, you will read about how to model more generalized motion using the [[183_notes:momentum_principle|momentum principle]].
+The motion of objects is not limited to [[183_notes:displacement_and_velocity|straight line motion]]. As you read earlier, [[183_notes:momentum_principle|forces can change the momentum of objects]] (including the direction of that momentum). These interactions can produce [[183_notes:localg|projectile motion]], [[183_notes:ucm|circular motion]], [[183_notes:springmotion|oscillations]], or more generalized trajectories. **In these notes, you will read about how to model more generalized motion using the [[183_notes:momentum_principle|momentum principle]].**
+==== Lecture Video ====
+{{youtube>wm2NbUDoAV0?large}}
 ==== The Derivative form of the Momentum Principle ====
@@ Line 59: / Line 65: @@
 $$\dfrac{d \hat{p}}{d t} = \dfrac{|\vec{v}|}{R} \hat{n}$$
-[{{183_notes:mi3e_05-026.jpg?250|The perpendicular component of the net force points inward towards the turn.}}]
+[{{183_notes:mi3e_05-026.png?250|The perpendicular component of the net force points inward towards the turn.}}]
 where the unit vector, $\hat{n}$, always points inward towards the turn. For more general trajectories, the value of $R$ is the radius of curvature of the arc, that is, it is the radius of the circle that has exactly the same curvature at the location of interest.
@@ Line 69: / Line 75: @@
 ==== 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 ($\vec{a}_{c}$)]]. This 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 [[http://en.wikipedia.org/wiki/Acceleration#Tangential_and_centripetal_acceleration| 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.
 $$\vec{F}_{net} = \vec{F}_{\parallel} + \vec{F}_{\perp}$$
 $$\vec{F}_{\parallel}  = m\vec{a}_{t} = m{a}_{t}\hat{p} \qquad \vec{F}_{\perp} = m\vec{a}_{c} = m{a}_{c}\hat{n}$$
-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.
+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 (or opposite it in the case of negative acceleration). 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.
+==== Video of Bowling Ball Moving in a Circle ====
+In this video a bowling ball is forced to move in a circle by being struck with a sledgehammer. This video was originally collected by [[http://paer.rutgers.edu|Eugenia Etkina and David Brookes]].
+{{183_notes:bowlingball.mp4}}
+==== Examples ====
+  * [[:183_notes:examples:videoswk6|Video Example: Change in momentum (parallel and perpendicular) of an orbit]]