Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.3_volume_integrals [2022/01/04 19:30] – [5.3.3 The force of an extended object (Earth) on a point particle (professor)] 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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   &= - \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}   &= - \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.1641321024.txt.gz · Last modified: 2022/01/04 19:30 by abril