Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.3_volume_integrals [2022/01/04 19:15] – [5.3.2 Change of variables] abrilbook:chap5:5.3_volume_integrals [2022/01/04 20:29] (current) – abril
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-FIXME draft with missing figures and references :!: 
- 
 ===== 5.3  Volume integrals — A professor falling through Earth  ===== ===== 5.3  Volume integrals — A professor falling through Earth  =====
 <WRAP #section_VolumeMassDensity></WRAP> <WRAP #section_VolumeMassDensity></WRAP>
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-<WRAP box round>**Example 5.2** <wrap em>Volume of a sphere</wrap> \\ +<WRAP box round #bsp_VolumeIntegral>**Example 5.2** <wrap em>Volume of a sphere</wrap> \\ 
 The volume of a three-dimensional sphere $S$ with center at the origin and radius $R$ is The volume of a three-dimensional sphere $S$ with center at the origin and radius $R$ is
 \begin{align*}  \begin{align*} 
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 </WRAP> </WRAP>
  
-<WRAP box round>**Example 5.3** <wrap em>Integration volumes</wrap> \\ +<WRAP box round #bsp_IntegrationVolumes>**Example 5.3** <wrap em>Integration volumes</wrap> \\ 
  
 **a)** //polar coordinates// **a)** //polar coordinates//
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 a physics professor and its environment. a physics professor and its environment.
 In the absence of interaction with other matter the professor will freely fall towards the center of Earth, In the absence of interaction with other matter the professor will freely fall towards the center of Earth,
-accelerated by a force that arises as sum of the mass elements constituting Earth (see\cref{fig:FallingThroughEarth}).+accelerated by a force that arises as sum of the mass elements constituting Earth (see [[#fig_FallingThroughEarth |Figure 5.10]]).
 For the professor of mass $m$ at position $\mathbf q_P$ and the mass element at position $\mathbf q_e$ this force amounts to For the professor of mass $m$ at position $\mathbf q_P$ and the mass element at position $\mathbf q_e$ this force amounts to
 $\mathbf F(\mathbf q_P, \mathbf q_e) = -\nabla ( m \, \rho(\mathbf q_e) \, G ) / \left\lvert \mathbf q_P - \mathbf q_e \right\rvert$. $\mathbf F(\mathbf q_P, \mathbf q_e) = -\nabla ( m \, \rho(\mathbf q_e) \, G ) / \left\lvert \mathbf q_P - \mathbf q_e \right\rvert$.
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 Then, the force on the professor takes the form Then, the force on the professor takes the form
  
-<WRAP right id=fig_FallingThroughEarth> +<WRAP 120pt right #fig_FallingThroughEarth> 
-{{./Sketch/FallingThroughEarth.png}} +{{FallingThroughEarth.png}} 
- +Figure 5.10: Initially positioned at the upper right (yellow), the professor will fall down (red),
-Initially positioned at the upper right (yellow), the professor will fall down (red),+
 and eventually pop out at the other side and return (green). and eventually pop out at the other side and return (green).
 </WRAP> </WRAP>
  
-\begin{align} \label{eq:professor-totForce}+<wrap #eq_professor-totForce></wrap> 
 +\begin{align}
   \mathbf F_{\text{tot}}   \mathbf F_{\text{tot}}
-  &= - \int_{\mathbb R^3}  \mathrm{d}^3 q \: \nabla \frac{ m \, \rho(\mathbf q_e) \, G }{ \left\lvert \mathbf q_P - \mathbf q_e \right\rvert }+  &= - \int_{\mathbb R^3}  \mathrm{d}^3 q \: \nabla \frac{ m \, \rho(\mathbf q_e) \, G }{ \left\lvert \mathbf q_P - \mathbf q_e \right\rvert } \tag{5.3.1}
   \\   \\
-  &= -   m \, \rho \, G \; \nabla  \int_{\text{Earth}}  \mathrm{d}^3 q \; \frac{1}{ \sqrt{ q_P^2 + q_e^2 - 2\, q_P \, q_e \, \cos\theta } }+  &= -   m \, \rho \, G \; \nabla  \int_{\text{Earth}}  \mathrm{d}^3 q \; \frac{1}{ \sqrt{ q_P^2 + q_e^2 - 2\, q_P \, q_e \, \cos\theta } } \tag{5.3.2}
 \end{align} \end{align}
 where $\theta$ is the angle between the two vectors $\left\lvert \mathbf q_P \right\rvert$ and $\left\lvert \mathbf q_e \right\rvert$, where $\theta$ is the angle between the two vectors $\left\lvert \mathbf q_P \right\rvert$ and $\left\lvert \mathbf q_e \right\rvert$,
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 $g = MG/R = 4\pi\,\rho\,R^2\,G/3$. $g = MG/R = 4\pi\,\rho\,R^2\,G/3$.
 The professor moves under the influence of a //harmonic// central force, The professor moves under the influence of a //harmonic// central force,
-as studied in\cref{quest:CoordTrafos,quest:ODE-12,quest:Conservation-08}! +as studied in 
-After a while (\cf\cref{quest:volIntegral-professor}) he reappears at the very same spot where he started,\+[[book:chap4:4.6_the_center_of_mass_cm_inertial_frame #quest_CoordTrafos |Problem 4.19]], 
 +[[book:chap4:4.9_solving_odes_by_coordinate_transformations #quest_ODE-12 |Problem 4.21]] and 
 +[[book:chap4:4.10_problems #quest_Conservation-08 |Problem 4.30]]! 
 + 
 +After a while (cf [[book:chap5:5.7_problems #quest_volIntegral-professor |Problem 5.12]]) he reappears at the very same spot where he started,
 except that Earth moved on while he was under way. except that Earth moved on while he was under way.
  
  
 ==== 5.3.4 Self Test ==== ==== 5.3.4 Self Test ====
- 
-----  
  
 <wrap #quest_volIntegral-parallelogram > Problem 5.5: </wrap>** Area of a parallelogram ** <wrap #quest_volIntegral-parallelogram > Problem 5.5: </wrap>** Area of a parallelogram **
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 describes a sphere of radius  $R$. describes a sphere of radius  $R$.
 The volume  $V$ of a solid of revolution are given by the integral The volume  $V$ of a solid of revolution are given by the integral
-\begin{align}\label{eq:revolutionSolidVolume} + 
-    V = \pi \; \int \mathrm{d} x \: \left( f(x) \right)^2+<wrap #eq_revolutionSolidVolume></wrap> 
 +\begin{align} 
 +    V = \pi \; \int \mathrm{d} x \: \left( f(x) \right)^2 \tag{5.3.3}
 \end{align} \end{align}
  
-  -  Sketch the function  $f(x) = \sqrt{R^2 - x^2}$ +  -  Sketch the function  $f(x) = \sqrt{R^2 - x^2}$ and verify that the solid of revolution is indeed a sphere. 
-and verify that the solid of revolution is indeed a sphere. +  -  Determine the volume of the sphere based on the given equation Compare you calculation and the result to the calculation given in [[#bsp_VolumeIntegral |Example 5.2]]. 
-  -  Determine the volume of the sphere based on the given equation. +  -  Show that the volume integral for a solid of revolution provides [[#eq_revolutionSolidVolume |Equation 5.3.3]]  when one adopts cylindrical coordinates.
-Compare you calculation and the result to the calculation given in \Example{VolumeIntegral}. +
-  -  Show that the volume integral for a solid of revolution provides  +
-\cref{eq:revolutionSolidVolume}  +
-when one adopts cylindrical coordinates.+
  
    
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 \\ \\
 Determine the Jacobi matrix and its determinant for the transformation Determine the Jacobi matrix and its determinant for the transformation
-from Cartesian to spherical coordinates, \cf\Example{IntegrationVolumes}c).+from Cartesian to spherical coordinates, cf [[#bsp_IntegrationVolumes |Example 5.3 c)]].
  
  
 ~~DISCUSSION~~  ~~DISCUSSION~~ 
  
book/chap5/5.3_volume_integrals.1641320158.txt.gz · Last modified: 2022/01/04 19:15 by abril