Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap1:1.1_basic_notions_of_mechanics [2021/10/07 06:01] – created jvbook:chap1:1.1_basic_notions_of_mechanics [2022/04/01 19:28] (current) – jv
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 +[[basics| 1. Basic Principles]]
 +  * **1.1 Basic notions of mechanics**
 +  * [[ 1.2 Dimensional analysis ]]
 +  * [[ 1.3 Order-of-magnitude guesses ]]
 +  * [[ 1.4 Problems ]]
 +  * [[ 1.5  Further reading ]]
 +
 +----
 +
 ===== 1.1 Basic notions of mechanics ===== ===== 1.1 Basic notions of mechanics =====
  
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 <wrap lo>**Remark.**  <wrap lo>**Remark.** 
-The arrows indicate here that $\mathbf x_i$ describes a position in space. +Bold-face symbols indicate here that $\mathbf x_i$ describes a position in space. 
-For a $D$-dimensional space one needs $D$ numbers((Strictly speaking we do not only need numbers, but must also indicate the adopted units.))+For a $D$-dimensional space one needs $D$ numbers((Strictly speaking we do not only need numbers,  
 +but must also indicate the adopted units.))
 to specify the position, to specify the position,
 and $\mathbf x_i$ may be thought of as a vector in $\mathbb{R}^D$. and $\mathbf x_i$ may be thought of as a vector in $\mathbb{R}^D$.
-We say that $\mathbf x_i$ is a $D$-vector. +We say that $\mathbf x_i$ is a $D$-vector. 
 +In [[book:chap2:forcestorques|Chapter 2]] we will take a closer look at vectors and their properties.
 </wrap> </wrap>
  
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 the latter will also be denoted as $\dot{\mathbf x}$. the latter will also be denoted as $\dot{\mathbf x}$.
 </wrap> </wrap>
 +
 +<wrap lo>**Remark.** 
 +In hand writing vectors are commonly denoted by an arrow, i.e., $\vec x$
 +rather than $\mathbf x$. 
 +</wrap>
 +
  
 <WRAP box round>**Example 1.1** <wrap hi>A piece of chalk</wrap> \\  <WRAP box round>**Example 1.1** <wrap hi>A piece of chalk</wrap> \\ 
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 In this model we have $N=2$ and $D=3$. In this model we have $N=2$ and $D=3$.
 </WRAP> </WRAP>
 +
 <WRAP box round>**Definition 1.2** <wrap em>Degrees of Freedom (DOF)</wrap> \\  <WRAP box round>**Definition 1.2** <wrap em>Degrees of Freedom (DOF)</wrap> \\ 
 A system with $N$ particles whose positions are described by $D$-vectors A system with $N$ particles whose positions are described by $D$-vectors
 has $D\,N$ //degrees of freedom (DOF)//. has $D\,N$ //degrees of freedom (DOF)//.
 </WRAP> </WRAP>
 +
 <wrap lo>**Remark.**  <wrap lo>**Remark.** 
 Note that according to this definition the number of DOF is a property of the model. Note that according to this definition the number of DOF is a property of the model.
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 However, the length of the piece of chalk does not change. However, the length of the piece of chalk does not change.
 Therefore, one can find an alternative description that will only evolve $5$ DOF. Therefore, one can find an alternative description that will only evolve $5$ DOF.
-(We will come back to this in due time.) +(We will come back to this in due time.) 
- +
 </wrap> </wrap>
 +
 <WRAP box round>**Definition 1.3** <wrap em>State Vector</wrap> \\  <WRAP box round>**Definition 1.3** <wrap em>State Vector</wrap> \\ 
 The position of all particles can be written in a single //state vector//, The position of all particles can be written in a single //state vector//,
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 Its components are called coordinates. Its components are called coordinates.
 </WRAP> </WRAP>
 +
 <wrap lo>**Remark.**  <wrap lo>**Remark.** 
 For a system with $N$ particles whose positions are specified by $D$-dimensional vectors, For a system with $N$ particles whose positions are specified by $D$-dimensional vectors,
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 The vector $\mathbf q$ has DOF number of entries, The vector $\mathbf q$ has DOF number of entries,
 and hence $\mathbf q \in \mathbb{R}^{DN}$. and hence $\mathbf q \in \mathbb{R}^{DN}$.
-  
 </wrap> </wrap>
 +
 <wrap lo>**Remark.**  <wrap lo>**Remark.** 
 The velocity associated to $\mathbf q$ will be denoted as The velocity associated to $\mathbf q$ will be denoted as
-$\dot{\mathbf q} = ( \dot{\mathbf x}_1, \dots , \dot{\mathbf x}_N )$. +$\dot{\mathbf q} = ( \dot{\mathbf x}_1, \dots , \dot{\mathbf x}_N )$. 
- +
 </wrap> </wrap>
 <WRAP box round>**Definition 1.4** <wrap em>Phase Vector</wrap> \\  <WRAP box round>**Definition 1.4** <wrap em>Phase Vector</wrap> \\ 
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 </WRAP> </WRAP>
  
-<WRAP box round>**Definition 1.6** <wrap em>Initial Conditions (IC)</wrap> \\ +<WRAP box round #Defi_IC>**Definition 1.6** <wrap em>Initial Conditions (IC)</wrap> \\ 
 equations of motion}} equations of motion}}
 ordinary differential equation}} ordinary differential equation}}
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 </wrap> </wrap>
  
-<WRAP 150px right> +<WRAP 150px right #fig_javelin-IC > 
-{{ en:book:chap01:jovelin-ic.png?direct&150  |}}+{{ book:chap1:jovelin-ic.png?direct&150  |}}
 <wrap lo>based on [[https://commons.wikimedia.org/wiki/File:JavelinElliott7435.jpg|Atalanta, creativecommons]],</wrap>\\ <wrap lo>based on [[https://commons.wikimedia.org/wiki/File:JavelinElliott7435.jpg|Atalanta, creativecommons]],</wrap>\\
 <wrap>[[http://creativecommons.org/licenses/by-sa/3.0/|CC BY-SA 3.0]]</wrap> \\  <wrap>[[http://creativecommons.org/licenses/by-sa/3.0/|CC BY-SA 3.0]]</wrap> \\ 
-Initial conditions for throwing a javelin.  +Figure 1.1: Initial conditions for throwing a javelin, cf. [[#bsp_javelin-IC|Example 1.2]]. 
-<wrap hide>, cf. \Example{javelin-IC}. \label{figure:javelin-IC}}</wrap>+
 </WRAP> </WRAP>
  
-<WRAP box round>**Example 1.2** <wrap hi>Throwing a javelin</wrap> +<WRAP box round #bsp_javelin-IC >**Example 1.2** <wrap hi>Throwing a javelin</wrap>
-<wrap hide>label=example:javelin-IC</wrap>+
 The ICs for the flight of a javelin specify where it is released, $\mathbf x_0$, when it is thrown, The ICs for the flight of a javelin specify where it is released, $\mathbf x_0$, when it is thrown,
 the velocity $\mathbf v_0$ at that point of time, the velocity $\mathbf v_0$ at that point of time,
 and the orientation of the javelin. and the orientation of the javelin.
 In a good trial the initial orientation of the javelin is parallel to its initial velocity $\mathbf v_0$, In a good trial the initial orientation of the javelin is parallel to its initial velocity $\mathbf v_0$,
-as shown in the figure. <wrap hide>\cref{figure:javelin-IC}</wrap>+as shown in [[#javelin-IC|Figure 1.1]].
 </WRAP> </WRAP>
  
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 that one makes upon setting up the experiment. that one makes upon setting up the experiment.
 For instance  For instance 
-\\ +  * Beckham's banana kicks can only be understood when one accounts for the impact of air friction on the soccer ball. 
-Beckham's banana kicks can only be understood when one accounts for the impact of air friction on the soccer ball. +  * Air friction will not impact the trajectory of a small piece of talk that I throw into the dust bin. 
-\\ +
-Air friction will not impact the trajectory of a small piece of talk that I through into the dust bin. +
-\\+
 By adopting a clever choice of the parameterization the trajectory of the piece of chalk By adopting a clever choice of the parameterization the trajectory of the piece of chalk
 can be described in a setting with $5$ DOF. can be described in a setting with $5$ DOF.
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 <WRAP box round>**Example 1.6** <wrap hi>Physical Quantities</wrap> \\  <WRAP box round>**Example 1.6** <wrap hi>Physical Quantities</wrap> \\ 
-1. The mass, $M$, of a soccer ball can be fully characterized by a number and the unit kilogram (kg), e.g. $M \approx 0.4\text{kg}$.+1. The mass, $M$, of a soccer ball can be fully characterized by a number and the unit kilogram (kg), e.g. $M \approx 0.4\,\text{kg}$.
 \\ \\
 2. The length, $L$, of a piece of chalk can be fully characterized by a number and the unit meter (m), e.g. $L \approx 7 \times 10^{-2}\,\text{m}$. 2. The length, $L$, of a piece of chalk can be fully characterized by a number and the unit meter (m), e.g. $L \approx 7 \times 10^{-2}\,\text{m}$.
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 3. The duration, $T$, of a year can be characterized by a number and the unit second, e.g. $T \approx \pi\times \times 10^{7}\,\text{s}$. 3. The duration, $T$, of a year can be characterized by a number and the unit second, e.g. $T \approx \pi\times \times 10^{7}\,\text{s}$.
 \\ \\
-4. The speed, $v$, of a car can be fully characterized by a number and the unit, e.g. $v \approx 42\text{km/\hour}$.+4. The speed, $v$, of a car can be fully characterized by a number and the unit, e.g. $v \approx 42\,\text{km/h}$.
 \\ \\
 5. A position in a $D$-dimensional space can fully be characterized by $D$ numbers and the unit meter. 5. A position in a $D$-dimensional space can fully be characterized by $D$ numbers and the unit meter.
 \\ \\
-6. The velocity of a piece of chalk flying through the lecture hall can be characterized by three numbers and the unitm/s.+6. The velocity of a piece of chalk flying through the lecture hall can be characterized by three numbers and the unit m/s.
 However, one is missing information in that case about its rotation. However, one is missing information in that case about its rotation.
 </WRAP> </WRAP>
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 A transparent way to do this for the speed of the car in the example above is by multiplications with one A transparent way to do this for the speed of the car in the example above is by multiplications with one
 \begin{align*} \begin{align*}
-    v = 72 \frac{\text{km}}{\text{\hour}} \: \frac{ 1\text{\hour} }{3.6 \times 10^{3}\,\text{s}} \; \frac{ 1 \times 10^{3}\,\text{m} }{1\text{km}}+    v = 72 \frac{\text{km}}{\text{h}} \: \frac{ 1\,\text{h} }{3.6 \times 10^{3}\,\text{s}} \; \frac{ 1 \times 10^{3}\,\text{m} }{1\,\text{km}}
     = \frac{72}{3.6} \text{m/s}     = \frac{72}{3.6} \text{m/s}
-    = 20\text{m/s}+    = 20\,\text{m/s}
 \end{align*} \end{align*}
 </WRAP> </WRAP>
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      *  a colloquium talk at our Physics Department must not run take longer than a micro-century,      *  a colloquium talk at our Physics Department must not run take longer than a micro-century,
      *  a generous thumb-width amounts to one atto-parsec.      *  a generous thumb-width amounts to one atto-parsec.
-  -  The Physics Handbook of \citet{NordlingOesterman2006} defines a beard-second, i.e., the length an average beard grows in one second, as $10\text{nm}$. In contrast, //Google Calculator// uses a value of only $5\text{nm}$. I prefer the one where the synodic period of the moon amounts to a beard-inch. Which one will that be? +  -  The  //Physics Handbook of Nordling and Österman (2006)//((Nordling, C., and J. Österman, 2006, Physics Handbook for Science and Engineering (Studentlitteratur, Lund), 8 edition, ISBN 91-44-04453-4, quoted after [[https://en.wikipedia.org/wiki/List_of_humorous_units_of_measurement|Wikipedia’s List of humorous units of measurement]], accessed on 5 May 2020.)) defines a beard-second, i.e., the length an average beard grows in one second, as $10\,\text{nm}$. In contrast, //Google Calculator// uses a value of only $5\,\text{nm}.$ I prefer the one where the synodic period of the moon amounts to a beard-inch. Which one will that be? 
-  -  In the [[https://en.wikipedia.org/wiki/FFF_system|furlong–firkin–fortnight (FFF) unit system]] one furlong per fortnight amounts to the [[https://itotd.com/articles/2987/furlongs-per-fortnight/|speed of a tardy snail]] (1 centimeter per minute to a very good approximation), and one micro-fortnight was used as a delay for user input by some old-fashioned computers (it is equal to 1.2096\text{s}). Use this information to determine the length of one furlong.+  -  In the [[https://en.wikipedia.org/wiki/FFF_system|furlong–firkin–fortnight (FFF) unit system]] one furlong per fortnight amounts to the [[https://itotd.com/articles/2987/furlongs-per-fortnight/|speed of a tardy snail]] (1 centimeter per minute to a very good approximation), and one micro-fortnight was used as a delay for user input by some old-fashioned computers (it is equal to $1.2096\,\text{s}$). Use this information to determine the length of one furlong. 
  
 +~~DISCUSSION|Questions, Remarks, and Suggestions~~
  
-~~DISCUSSION~~ 
book/chap1/1.1_basic_notions_of_mechanics.1633579319.txt.gz · Last modified: 2021/10/07 06:01 by jv