Differences

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--- 183_notes:examples:a_meter_stick_on_the_ice [2014/11/16 21:20] – created pwirving
+++ 183_notes:examples:a_meter_stick_on_the_ice [2014/11/20 16:00] – pwirving
@@ Line 6: / Line 6: @@
 === Facts ===
+Mass of meter stick 300g
+Pull at end of meter stick at right angles to the stick: 6N
+Remember a meter stick is a meter long
@@ Line 13: / Line 16: @@
 === Lacking ===
+Rate of change of the center-of-mass speed $v_{CM}$?
+Rate of change of the angular speed $\omega$?
 === Approximations & Assumptions ===
+No friction due to ice
@@ Line 23: / Line 29: @@
 === Representations ===
+System: Stick
-{{course_planning:projects:mi3e_11-006.jpg?400}}
+Surroundings: Your hand (pulling); ice (negligible effect)
+{{183_projects:mi3e_11-050.jpg?300}}
+$\frac{d\vec{P}}{dt}$ = $\vec{F}_{net}$
+$\frac{d\vec{L}_{rot}}{dt}$ = $\vec{\tau}_{net,CM}$
-=== Solution ===
+$\tau = r_{A}Fsin \theta$
-Direction: At both locations, the direction of the translational angular momentum of the comet is in the -z direction (into the computer); determined by using the right-hand rule.
-At location 1:
-$\mid\vec{L}_{trans,Sun}\mid$ = $(8.77$ x $10^{10}m)(2.2$ x $10^{14}kg)(5.46$ x $10^4m/s)sin 90^{\circ}$
-$= 1.1$ x $10^{30}$ $kg \cdot m^2/s$
+=== Solution ===
+From the momentum principle:
+$d\vec{P}/dt = d(m\vec{v}_{CM})/dt = \vec{F}_{net}$
+$dv_{CM}/dt = (6N)/(0.3kg) = 20m/s^2$
-$\vec{L}_{trans,Sun}$ = $\langle{0, 0, -1.1 x 10^30}\rangle$ $kg \cdot m^2/s$
+Angular Momentum Principle about center of mass:
+$d\vec{L}_{rot}/dt = \vec{\tau}_{net,CM}$
-At location 2:
+Component into screen (-z direction):
-$\mid\vec{L}_{trans,Sun}\mid$ = $(1.19$ x $10^{12}m)(2.2$ x $10^{14}kg)(1.32$ x $10^4m/s)sin 17.81^{\circ}$
+$Id\omega/dt = (0.5m)(6N)sin90^{\circ} = 3N \cdot m$
-$= 1.1$ x $10^{30}$ $kg \cdot m^2/s$
+$d\omega/dt = (3N \cdot m)/[(0.3 kg \cdot m^2)/12] = 120 radians/s^2$
-$\vec{L}_{trans,Sun}$ = $\langle{0, 0, -1.1 x 10^30}\rangle$ $kg \cdot m^2/s$
+In vector terms, $d\vec{\omega}/dt$ points into the page, corresponding to the fact that the angular velocity points into the page and is increasing.