Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap1:1.3_order-of-magnitude_guesses [2021/10/07 06:14] – jvbook:chap1:1.3_order-of-magnitude_guesses [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 ]]
 +
 +----
 +
 ===== 1.3 Order-of-magnitude guesses ===== ===== 1.3 Order-of-magnitude guesses =====
  
 Many physical quantities take a value close to one Many physical quantities take a value close to one
-when they are expressed in their ``natural'' dimensionless units.+when they are expressed in their “natural” dimensionless units.
 When the choice is unique, then clearly it is also natural. When the choice is unique, then clearly it is also natural.
 Otherwise, the appropriate choice is a matter of experience. Otherwise, the appropriate choice is a matter of experience.
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 We demonstrate this based on a discussion of We demonstrate this based on a discussion of
  
 +<WRAP right 150px #fig_pendulum-omg >
 {{ book:chap1:pendulum_theta.png?direct&150|}} {{ book:chap1:pendulum_theta.png?direct&150|}}
-<wrap hide>\caption{Pendulum discussed in \Example{pendulum-omg} +Figure 1.3: Pendulum discussed in [[#bsp_pendulum-omg|Example 1.10]] 
-\label{figure:pendulum-omg}}</wrap> +</WRAP> 
-<WRAP box round>**Example 1.10** <wrap em>The period of a pendulum</wrap> \\ +<WRAP box round #bsp_pendulum-omg>**Example 1.10** <wrap em>The period of a pendulum</wrap> \\ 
 We consider a pendulum of mass $M$ attached at a stiff bar of negligible mass. We consider a pendulum of mass $M$ attached at a stiff bar of negligible mass.
 With this bar it is fixed to a pivot at a distance $L$ from the mass With this bar it is fixed to a pivot at a distance $L$ from the mass
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 In this example we make use of the fact that the bar has fixed length $L$, In this example we make use of the fact that the bar has fixed length $L$,
 and describe the position of the mass by the angle $\theta(t)$ and describe the position of the mass by the angle $\theta(t)$
-(see figure right).+(see [[#fig_pendulum-omg|Figure 1.3]]).
 \\ \\
-As discussed in \Example{pendulum-nodim} the dimensionless time unit for this problem is $\sqrt{L/g}$. +As discussed in [[book:chap1:1.2_dimensional_analysis#bsp_pendulum-nodim|Example 1.8]] the time unit for this problem is $\sqrt{L/g}$. 
-Hence we estimate that the period $T$ of the pendulum is of the order of+Hence, we estimate that the period $T$ of the pendulum is of the order of
 $ $
 T \simeq \sqrt{L/g}  T \simeq \sqrt{L/g} 
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 </WRAP> </WRAP>
  
-<WRAP box round>**Example 1.11** <wrap em>The speed of Tsunami waves</wrap> \\ +<WRAP box round #Tsunami-speed-omg>**Example 1.11** <wrap em>The speed of Tsunami waves</wrap> \\ 
 A Tsunami wave is a water wave that is generated by an earth quake or an underwater land slide. A Tsunami wave is a water wave that is generated by an earth quake or an underwater land slide.
 Typical wave lengths are of an order of magnitude $\lambda = 100\,\text{km}$. Typical wave lengths are of an order of magnitude $\lambda = 100\,\text{km}$.
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 \frac{L}{ v_{\text{Tsunami}} } \frac{L}{ v_{\text{Tsunami}} }
 \approx \frac{ 1 \times 10^{4}\,\text{km}}{ 700\,\text{km/h} } \approx \frac{ 1 \times 10^{4}\,\text{km}}{ 700\,\text{km/h} }
-= \frac{100}{7} \text{h}+= \frac{100}{7}\,\text{h}
 \approx 15\,\text{h} \approx 15\,\text{h}
 \] \]
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 T \approx \frac{\lambda}{ v_{\text{Tsunami}} } T \approx \frac{\lambda}{ v_{\text{Tsunami}} }
 = \frac{ \lambda }{  \sqrt{ g D } } = \frac{ \lambda }{  \sqrt{ g D } }
-= \frac{ 100\text{km} }{ 700\text{km/h} }+= \frac{ 100\,\text{km} }{ 700\,\text{km/h} }
 = \frac{1}{ 7\,\text{h} } = \frac{1}{ 7\,\text{h} }
 \approx 10\,\text{min} \approx 10\,\text{min}
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 ==== Self Test ==== ==== Self Test ====
- +<WRAP #quest_SIunits-21 > 
-Problem 1.5: <wrap hide>\label{quest:SIunits-21}</wrap>+Problem 1.5: 
 ** Printing the output of Phantom cameras ** ** Printing the output of Phantom cameras **
 \\ \\
-With a set of three phantom cameras one can simultaneously follow the motion of 100 particles+With a set of three phantom cameras one can simultaneously follow the motion of $100$ particles
 in a violent 3d turbulent flow. in a violent 3d turbulent flow.
-Data analysis of the images provides particle positions with a resolution of 25,000 frames per second. +Data analysis of the images provides particle positions with a resolution of $25\,000$ frames per second. 
-You follow the evolution for 20\text{minute}, +You follow the evolution for $20\,\text{minute}$, 
-print it double paged with 8 coordinates per line and 70 lines per page. +print it double paged with $8$ coordinates per line and $70$ lines per page. 
-A bookbinder makes 12\text{cm} thick books from every 1000 pages.+A bookbinder makes $12\,\text{cm}$ thick books from every $1000$ pages.
 You put these books into bookshelves with seven boards in each shelf. You put these books into bookshelves with seven boards in each shelf.
 How many meters of bookshelves will you need to store your data on paper? How many meters of bookshelves will you need to store your data on paper?
 +</WRAP>
 +
 +~~DISCUSSION|Questions, Remarks, and Suggestions~~
  
-~~DISCUSSION~~ 
book/chap1/1.3_order-of-magnitude_guesses.1633580080.txt.gz · Last modified: 2021/10/07 06:14 by jv