Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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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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 If the mass distribution of the body obeys reflection or rotation symmetry, If the mass distribution of the body obeys reflection or rotation symmetry,
 the axes of inertia are invariant under the symmetry transformation. the axes of inertia are invariant under the symmetry transformation.
- \manimpossiblecube 
 </wrap> </wrap>
  
  
 ==== 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.1641442568.txt.gz · Last modified: 2022/01/06 05:16 by abril