Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.4_center_of_mass_and_spin

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book:chap5:5.4_center_of_mass_and_spin [2022/01/06 05:16] – [5.4.2 Angular momentum and particle spin] abrilbook:chap5:5.4_center_of_mass_and_spin [2022/01/06 05:22] (current) – abril
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-FIXME draft with missing figures and references :!: 
- 
 ===== 5.4  Center of mass and spin of extended objects  ===== ===== 5.4  Center of mass and spin of extended objects  =====
 <WRAP #section_particleExtension></WRAP> <WRAP #section_particleExtension></WRAP>
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 (i.e., the Jacobi determinant of the transformation is one).  (i.e., the Jacobi determinant of the transformation is one). 
 The acceleration $\ddot{\boldsymbol q}(t)$ takes the form The acceleration $\ddot{\boldsymbol q}(t)$ takes the form
-<wrap #eq_body-Ftot}></wrap>+ 
 +<wrap #eq_body-Ftot></wrap> 
 \begin{align*}  \begin{align*} 
   \ddot{\boldsymbol q}(t) = \ddot{\boldsymbol q}(t) + \sum_{i=1}^3 r_i \: \ddot{\hat{\boldsymbol e}}_i(t)   \ddot{\boldsymbol q}(t) = \ddot{\boldsymbol q}(t) + \sum_{i=1}^3 r_i \: \ddot{\hat{\boldsymbol e}}_i(t)
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 ==== 5.4.3  Time evolution of angular momentum and particle spin  ==== ==== 5.4.3  Time evolution of angular momentum and particle spin  ====
-<WRAP id=ssection_spinEvolution />+<wrap #ssection_spinEvolution></wrap>
  
 The angular momentum $\mathbf L_{CM}$ of its center-of-mass motion The angular momentum $\mathbf L_{CM}$ of its center-of-mass motion
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     \dot{\boldsymbol S}     \dot{\boldsymbol S}
     = \mathbf M     = \mathbf M
-    = \int_{\text{body}} \mathrm{d}^3 r \;  \mathbf r \times \mathbf F( \mathbf Q + \mathbf r )+    = \int_{\text{body}} \mathrm{d}^3 r \;  \mathbf r \times \mathbf F( \mathbf Q + \mathbf r ) \tag{5.4.6}
 \end{align} \end{align}
 </WRAP> </WRAP>
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 where the gravitational acceleration $\mathbf g$ takes a constant value,  where the gravitational acceleration $\mathbf g$ takes a constant value, 
 forms a noticeable exception. forms a noticeable exception.
-<WRAP box round>**Theorem 5.3 <wrap hi> Spinning motion and gravity </wrap>** \\ + 
 +<WRAP box round>**Theorem 5.3 <wrap em> Spinning motion and gravity </wrap>** \\ 
 When an extended body moves subject to a spatially uniform acceleration $\mathbf g$, When an extended body moves subject to a spatially uniform acceleration $\mathbf g$,
 then its center of mass follows a free-flight parabola then its center of mass follows a free-flight parabola
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 //Proof.// //Proof.//
-The statement about the center-of-mass motion follows from \cref{eq:body-Ftot}.+The statement about the center-of-mass motion follows from [[#eq_body-Ftot |Equation 5.4.4]].
 Conservation of the spin is due to Conservation of the spin is due to
 \begin{align*}  \begin{align*} 
book/chap5/5.4_center_of_mass_and_spin.1641442602.txt.gz · Last modified: 2022/01/06 05:16 by abril