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--- book:chap1:1.2_dimensional_analysis [2021/10/07 06:04] – created jv
+++ book:chap1:1.2_dimensional_analysis [2022/04/01 19:28] (current) – jv
@@ Line 1: / Line 1: @@
+[[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 ]]
+----
 ===== 1.2 Dimensional analysis =====
@@ Line 8: / Line 17: @@
 The discussion of a more advanced formulation may appear as a homework problem later on on this course.
-<WRAP box round>**Theorem 1.1 <wrap hi>Buckingham-Pi-Theorem</wrap>** \\
+<WRAP box round #Buckingham-Pi>**Theorem 1.1 <wrap hi>Buckingham-Pi-Theorem</wrap>** \\
 A dynamics with $n$ parameters,
 where the positions $\mathbf q$ and the parameters involve the three units meter, seconds and kilogram,
@@ Line 14: / Line 23: @@
 with $n-3$ parameters,
 where the positions $\boldsymbol\xi$, velocities $\boldsymbol\zeta$,
-and parameters $\pi_j$ with $j \in \{ 1, \dots, n-1\}$
+and parameters $\pi_j$ with $j \in \{ 1, \dots, n-3\}$
 are given solely by numbers.
 </WRAP>
+<WRAP 200px right #fig_pendulum-nodim >
 {{ book:chap1:pendulum_xz.png?direct&200|}}
-<wrap hide>\caption{Pendulum discussed in \Example{pendulum-nodim}
+Figure 1.2: Pendulum discussed in [[#bsp_pendulum-nodim|Example 1.8]].
-\label{figure:pendulum-xz}}
+</WRAP>
-\end{marginfigure}
-</wrap>
+<WRAP box round #bsp_pendulum-nodim >**Example 1.8** <wrap em>Non-dimensionalization for a pendulum</wrap> \\
-<WRAP box round>**Example 1.8** <wrap em>Non-dimensionalization for a pendulum</wrap> \\
 Let $\mathbf x$ denote the position of a pendulum of mass $M$
 that is attached to a chord of length $L$
 and swinging in a gravitational field $\mathbf g$ of strength $g$
-(see the figure to the right).
+(see [[#fig_pendulum-nodim|Figure 1.2]]).
 \\
 The units of these quantities are
@@ Line 110: / Line 119: @@
 ==== Self Test ====
+<WRAP #quest_SIunits-30 >
-Problem 1.3: <wrap hide>\label{quest:SIunits-30}</wrap>
+Problem 1.3:
 ** Oscillation period of a particle attached to a spring **
 \\
-In a gravitational field with acceleration $g_{\text{Moon}}=1.6\text{m/s$^2$}$
+In a gravitational field with acceleration $g_{\text{Moon}}=1.6\,\text{m/s$^2$}$
-a particle of mass $M=100\text{g}$
+a particle of mass $M=100\,\text{g}$
-is hanging at a spring with spring constant $k=1.6\text{kg/s$^2$}$.
+is hanging at a spring with spring constant $k=1.6\,\text{kg/s$^2$}$.
 It oscillates with period $T$ when it is slightly pulled downwards and released.
 We describe the oscillation by the distance $x(t)$ from its rest position.
@@ Line 122: / Line 131: @@
   -  Determine the dimensionless distance $\xi(t)$,\\ and the associated dimensionless velocity$\zeta(t)$.
   -  Provide an order-of-estimate guess of the oscillation period $T$.
+</WRAP>
+----
-Problem 1.4:  <wrap hide>\label{quest:SIunits-31}</wrap>
+<WRAP #quest_SIunits-31 >
+Problem 1.4:
 ** Earth orbit around the sun **
 \\
-  -  Light travels with a speed of $c \approx 3 \times 10^{8}\,\text{m/s}$, and it takes 500\text{s} to travel from Sun to Earth. What is the Earth-Sun distance$D$, i.e., one Astronomical Unit (AU) in meters?
+  -  Light travels with a speed of $c \approx 3 \times 10^{8}\,\text{m/s}$, and it takes $500\,\text{s}$ to travel from Sun to Earth. What is the Earth-Sun distance$D$, i.e., one Astronomical Unit (AU) in meters?
   -  The period of the trajectory of the Earth around the Sun depends on $D$, on the mass $M = 2 \times 10^{30}\,\text{kg}$ of the sun, and on the gravitational constant $G = 6.7 \times 10^{-11}\,\text{m$^3$/kg s$^2$}$. Estimate, based on this information, how long it takes for the Earth to travel once around the sun.
   -  Express your estimate in terms of years.  The estimate of (b) is of order one, but still off by a considerable factor. Do you recognize the numerical value of this factor?
   -  Upon discussing the trajectory $\mathbf x(t)$ of planets around the sun later on in this course, we will introduce dimensionless positions of the planets $\boldsymbol\xi(t) = \mathbf x(t) / L = ( x_1(t)/L, x_2(t)/L, x_3(t)/L)$. How would you define the associated dimensionless velocities?
+</WRAP>
+~~DISCUSSION|Questions, Remarks, and Suggestions~~
-~~DISCUSSION~~