Forest UCM RBM - Revision history

Foretony: /* Two principal moments are the same value (Free precession) */

2014-12-11T15:56:33Z

Two principal moments are the same value (Free precession)

2014-12-11T15:54:40Z

Two principal moments are the same value (Free precession)

2014-12-11T15:54:22Z

Two principal moments are the same value (Free precession)

2014-12-11T15:53:36Z

Two principal moments are the same value (Free precession)

2014-12-11T15:51:49Z

Two principal moments are the same value (Free precession)

2014-12-11T15:51:32Z

Two principal moments are the same value (Free precession)

2014-12-11T15:51:10Z

Two principal moments are the same value (Free precession)

2014-12-11T15:50:41Z

Two principal moments are the same value (Free precession)

2014-12-11T15:47:13Z

Two principal moments are the same value (Free precession)

2014-12-11T15:44:32Z

Two principal moments are the same value (Free precession)

@@ Line 793: / Line 793: @@
 : <math>\vec \omega =\omega_0  \sin (\Omega_b t)  \hat e_1 - \omega_0  \cos (\Omega_b t) \hat e_2 + \omega_3 \hat e_3 </math>
 ;Interpretation
@@ Line 808: / Line 810: @@
 ;Observation
 : The Body frame (the reference frame attached to the spinning top) sees  <math>\vec \omega</math> and <math>\vec L</math> precess about <math>\hat e_3</math> at an angular velocity of <math>\Omega_b</math>
 [[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]

@@ Line 807: / Line 807: @@
 ;Observation
-: The Body frame (the reference frame attached to the spinning top) sees  <math>\vec \omega</math> and <math>\vec L</math> precess about <<math>\hat e_3</math> at an angular velocity of <math>\Omega_b</math>
+: The Body frame (the reference frame attached to the spinning top) sees  <math>\vec \omega</math> and <math>\vec L</math> precess about <math>\hat e_3</math> at an angular velocity of <math>\Omega_b</math>
 [[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]

@@ Line 807: / Line 807: @@
 ;Observation
-: The Body frame (the reference frame attached to the spinning top) sees  <math>\vec \omega</math> and <math>\vec L</math> precess about <math>\hat e_3</math>
+: The Body frame (the reference frame attached to the spinning top) sees  <math>\vec \omega</math> and <math>\vec L</math> precess about <<math>\hat e_3</math> at an angular velocity of <math>\Omega_b</math>
 [[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]

← Older revision		Revision as of 15:53, 11 December 2014
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−		+	;Observation
		+	: The Body frame (the reference frame attached to the spinning top) sees <math>\vec \omega</math> and <math>\vec L</math> precess about <math>\hat e_3</math>


	[[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]		[[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]

← Older revision		Revision as of 15:51, 11 December 2014
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	: <math>\vec L = \tilde I \vec \omega = \lambda_1 \omega_1 \hat e_1 + \lambda_1 \omega_2 \hat e_2 + \lambda_3 \omega_3 \hat e_3</math>		: <math>\vec L = \tilde I \vec \omega = \lambda_1 \omega_1 \hat e_1 + \lambda_1 \omega_2 \hat e_2 + \lambda_3 \omega_3 \hat e_3</math>
−	:: <math>\lambda_1 \omega_0 \sin (\Omega_b t) \hat e_1 - \lambda_1 \omega_0 \cos (\Omega_b t) \hat e_2 + \lambda_3 \omega_3 \hat e_3</math>	+	:: <math>=\lambda_1 \omega_0 \sin (\Omega_b t) \hat e_1 - \lambda_1 \omega_0 \cos (\Omega_b t) \hat e_2 + \lambda_3 \omega_3 \hat e_3</math>

@@ Line 802: / Line 802: @@
 The angular momentum (<math>\vec L</math> ) in the space frame is given by
-: <math>\vec L = \tilde I \vec \omega = \lamdba_1 \omega_1 \hat e_1 +  \lambda_1 \omega_2 \hat e_2 +  \lambda_3 \omega_3 \hat e_3</math>
+: <math>\vec L = \tilde I \vec \omega = \lambda_1 \omega_1 \hat e_1 +  \lambda_1 \omega_2 \hat e_2 +  \lambda_3 \omega_3 \hat e_3</math>
 :: <math>\lambda_1 \omega_0  \sin (\Omega_b t)  \hat e_1 -  \lambda_1 \omega_0  \cos (\Omega_b t) \hat e_2 +  \lambda_3 \omega_3 \hat e_3</math>

@@ Line 783: / Line 783: @@
 Then
 : <math>\eta = \eta_0 e^{-i \Omega_b t}= \omega_0 e^{-i \Omega_b t} </math>
-: <math> \omega_1 + i \omega_2 =\omega_0 \left ( \sin (\Omega_b t) +i \cos (\Omega_b t)\right )  </math>
+: <math> \omega_1 + i \omega_2 =\omega_0 \left ( \sin (\Omega_b t) -i \cos (\Omega_b t)\right )  </math>
-: <math>\omega_1 = \omega_0  \sin (\Omega_b t)  \;\;\;\;\omega_2 = \omega_0  \cos (\Omega_b t)</math>
+: <math>\omega_1 = \omega_0  \sin (\Omega_b t)  \;\;\;\;\omega_2 = -\omega_0  \cos (\Omega_b t)</math>
 <math>\Rightarrow</math>
-: <math>\vec \omega =\omega_0  \sin (\Omega_b t)  \hat e_1 + \omega_0  \cos (\Omega_b t) \hat e_2 + \omega_3 \hat e_3 </math>
+: <math>\vec \omega =\omega_0  \sin (\Omega_b t)  \hat e_1 - \omega_0  \cos (\Omega_b t) \hat e_2 + \omega_3 \hat e_3 </math>
@@ Line 799: / Line 799: @@
 ;Body cone
 :The Body cone is defined as the orbital path taken by the top as it rotates about the <math>\hat e_3</math> axis with the angular velocity <math>\Omega_b</math>.  The shape forms a cone relative to the rotation point of the top (the top's tip)
 [[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]

← Older revision		Revision as of 15:47, 11 December 2014
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	:the two components, <math>\omega_1</math> and <math>\omega_2</math>, of the rigid body's angular velocity <math>\vec \omega</math> are rotating with an angular velocity <math>\Omega_b</math> about the <math>\hat e_3</math> axis as viewed in the Body reference frame.		:the two components, <math>\omega_1</math> and <math>\omega_2</math>, of the rigid body's angular velocity <math>\vec \omega</math> are rotating with an angular velocity <math>\Omega_b</math> about the <math>\hat e_3</math> axis as viewed in the Body reference frame.

		+	;Body cone
		+	:The Body cone is defined as the orbital path taken by the top as it rotates about the <math>\hat e_3</math> axis with the angular velocity <math>\Omega_b</math>. The shape forms a cone relative to the rotation point of the top (the top's tip)

	[[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]		[[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]

@@ Line 795: / Line 795: @@
 ;Interpretation
-:the two components, <math>\omega_1</math> and <math>\omega_2</math>, of the rigid body's angular velocity <math>\vec \omega</math> are rotating with an angular velocity <math>\Omega_b</math> about the <math>\hat e_3</math> axis relative as viewed in the Body reference frame.
+:the two components, <math>\omega_1</math> and <math>\omega_2</math>, of the rigid body's angular velocity <math>\vec \omega</math> are rotating with an angular velocity <math>\Omega_b</math> about the <math>\hat e_3</math> axis as viewed in the Body reference frame.
 [[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]