Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap1:1.1_basic_notions_of_mechanics [2021/10/25 22:45] – fix cross references 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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 </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 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. $v \approx 42\text{km/h}$.+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.
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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{h}} \: \frac{ 1\text{h} }{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 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? +  -  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|Questions, Remarks, and Suggestions~~
  
book/chap1/1.1_basic_notions_of_mechanics.1635194741.txt.gz · Last modified: 2021/10/25 22:45 by jv