book:chap1:1.2_dimensional_analysis
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book:chap1:1.2_dimensional_analysis [2021/10/07 09:31] – [Self Test] jv | book:chap1:1.2_dimensional_analysis [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 ]] | ||
+ | |||
+ | ---- | ||
+ | |||
===== 1.2 Dimensional analysis ===== | ===== 1.2 Dimensional analysis ===== | ||
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The discussion of a more advanced formulation may appear as a homework problem later on on this course. | The discussion of a more advanced formulation may appear as a homework problem later on on this course. | ||
- | <WRAP box round> | + | <WRAP box round # |
A dynamics with $n$ parameters, | A dynamics with $n$ parameters, | ||
where the positions $\mathbf q$ and the parameters involve the three units meter, seconds and kilogram, | where the positions $\mathbf q$ and the parameters involve the three units meter, seconds and kilogram, | ||
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with $n-3$ parameters, | with $n-3$ parameters, | ||
where the positions $\boldsymbol\xi$, | where the positions $\boldsymbol\xi$, | ||
- | 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. | are given solely by numbers. | ||
</ | </ | ||
+ | <WRAP 200px right # | ||
{{ book: | {{ book: | ||
- | <wrap hide> | + | Figure 1.2: Pendulum discussed in [[# |
- | \label{figure: | + | </WRAP> |
- | \end{marginfigure} | + | |
- | </wrap> | + | <WRAP box round # |
- | <WRAP box round> | + | |
Let $\mathbf x$ denote the position of a pendulum of mass $M$ | Let $\mathbf x$ denote the position of a pendulum of mass $M$ | ||
that is attached to a chord of length $L$ | that is attached to a chord of length $L$ | ||
and swinging in a gravitational field $\mathbf g$ of strength $g$ | and swinging in a gravitational field $\mathbf g$ of strength $g$ | ||
- | (see the figure to the right). | + | (see [[# |
\\ | \\ | ||
The units of these quantities are | The units of these quantities are | ||
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==== Self Test ==== | ==== Self Test ==== | ||
- | + | <WRAP # | |
- | Problem 1.3: <wrap hide> | + | Problem 1.3: |
** Oscillation period of a particle attached to a spring ** | ** Oscillation period of a particle attached to a spring ** | ||
\\ | \\ | ||
- | In a gravitational field with acceleration $g_{\text{Moon}}=1.6\text{m/ | + | In a gravitational field with acceleration $g_{\text{Moon}}=1.6\,\text{m/ |
- | 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/ | + | is hanging at a spring with spring constant $k=1.6\,\text{kg/ |
It oscillates with period $T$ when it is slightly pulled downwards and released. | 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. | We describe the oscillation by the distance $x(t)$ from its rest position. | ||
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- Determine the dimensionless distance $\xi(t)$,\\ and the associated dimensionless velocity$\zeta(t)$. | - Determine the dimensionless distance $\xi(t)$,\\ and the associated dimensionless velocity$\zeta(t)$. | ||
- Provide an order-of-estimate guess of the oscillation period $T$. | - Provide an order-of-estimate guess of the oscillation period $T$. | ||
+ | </ | ||
- | + | ---- | |
- | Problem 1.4: | + | <WRAP # |
+ | Problem 1.4: | ||
** Earth orbit around the sun ** | ** Earth orbit around the sun ** | ||
\\ | \\ | ||
- | - Light travels with a speed of $c \approx 3 \times 10^{8}\, | + | - Light travels with a speed of $c \approx 3 \times 10^{8}\, |
- The period of the trajectory of the Earth around the Sun depends on $D$, on the mass $M = 2 \times 10^{30}\, | - The period of the trajectory of the Earth around the Sun depends on $D$, on the mass $M = 2 \times 10^{30}\, | ||
- Express your estimate in terms of years. | - Express your estimate in terms of years. | ||
- 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? | - 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? | ||
+ | </ | ||
+ | |||
~~DISCUSSION|Questions, | ~~DISCUSSION|Questions, | ||
+ |
book/chap1/1.2_dimensional_analysis.1633591886.txt.gz · Last modified: 2021/10/07 09:31 by jv