16.B: Algunas ecuaciones y constantes
- Page ID
- 129687
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Constantes Físicas
| Nombre | Símbolo | Valor |
|---|---|---|
| Velocidad de la luz | \(c\) | \(3.00 \cdot 10^8 m/s\) |
| Cargo elemental | \(e\) | \(1.60 \cdot 10^{-19} C\) |
|
Masa de electrones |
\(m_e\) | \(9.11 \cdot 10^{-31} kg = 0.511 MeV/c^2\) |
| Masa protónica | \(m_p\) | \(1.67 \cdot 10^{-27} kg = 938 MeV/c^2\) |
| Constante gravitacional | \(G\) | \(6.67 \cdot 10^{-11} N \cdot m^2 /kg^2\) |
| Aceleración gravitacional | \(g\) | \(9.81 m/s^2\) |
| Constante de Boltzmann | \(k_B\) | \(1.38 \cdot 10^{-23} J/K\) |
| Constante de Planck |
\(h\) \(\hbar = h /2 \pi\) |
\(6.63 \cdot 10^{-34} J \cdot s\) \(1.05 \cdot 10^{-34} J \cdot s\) |
Momentos de inercia
| Objeto | Momento de inercia |
|---|---|
| Stick delgado (largo L) | \(\frac{1}{12} M L^2\) |
| Anillo de cilindro hueco (radio R) | \(M R^2\) |
| Disco o cilindro sólido (radio R) | \(\frac{1}{2} M R^2\) |
| Esfera hueca (radio R) | \(\frac{2}{3} M R^2\) |
| Esfera sólida (radio R) | \(\frac{2}{5} M R^2\) |
| Rectángulo (tamaño\(a \times b\)), eje perpendicular | \(\frac{1}{12} M (a^2 + b^2)\) |
| Rectángulo (tamaño\(a \times b\)), eje paralelo al lado b | \(\frac{1}{12} M a^2\) |
Objetos del Sistema Solar
| Sun | Tierra | Luna | |
|---|---|---|---|
| Masa (kg) | \(1.99 \cdot 10^{30}\) | \(5.97 \cdot 10^{24}\) | \(7.35 \cdot 10^{22}\) |
| Radio medio (m) | \(6.96 \cdot 10^{8}\) | \(6.37 \cdot 10^{6}\) | \(1.74 \cdot 10^{6}\) |
| Periodo (s) orbital |
\(6 \cdot 10^{15}\) (200 Mi) |
\(3.16 \cdot 10^{7}\) (365.25 días) |
\(2.36 \cdot 10^{6}\) (27.3 días) |
| Radio orbital medio (m) | \(2.6 \cdot 10^{20}\) | \(1.50 \cdot 10^{11}\) | \(3.85 \cdot 10^{8}\) |
| Densidad media (kg/m 3) | \(1.4 \cdot 10^{3}\) | \(5.5 \cdot 10^{3}\) | \(3.3 \cdot 10^{3}\) |
| Nombre | Símbolo | Radio ecuatorial | Masa | Radio medio de la órbita | Periodo orbital | Inclinación | Excentricidad orbital | Periodo de rotación | Lunas confirmadas | Inclinación axial |
|---|---|---|---|---|---|---|---|---|---|---|
| Mercurio |  |
0.382 | 0.06 | 0.39 | 0.24 | 3.38 | 0.206 | 58.64 | 0 | 0.04 |
| Venus |  |
0.949 | 0.82 | 0.72 | 0.62 | 3.86 | 0.007 | -243.02 | 0 | 177.36 |
| Tierra |  |
1 | 1 | 1 | 1 | 7.25 | 0.017 | 1 | 1 | 23.44 |
| Luna |  |
0.272 | 0.0123 | 384399 | 27.32158 | 18.29-28.58 | 0.0549 | 27.32158 | 0 | 6.68 |
| Marte |  |
0.532 | 0.107 | 1.52 | 1.88 | 5.65 | 0.093 | 1.03 | 2 | 25.19 |
| Ceres | 0.0742 | 0.00016 | 2.766 | 4.599 | 10.59 | 0.08 | 0.3781 | 0 | 4 | |
| Júpiter |  |
11.209 | 317.8 | 5.2 | 11.86 | 6.09 | 0.048 | 0.41 | 69 | 3.13 |
| Io | 0.285 | 0.015 | 421600 | 1.769 | 0.04 | 0.0041 | 1.769 | 0 | 0 | |
| Europa | 0.246 | 0.008 | 670900 | 3.551 | 0.47 | 0.009 | 3.551 | 0 | 0 | |
| Ganímedes | 0.423 | 0.025 | 1070400 | 7.155 | 1.85 | 0.0013 | 7.155 | 0 | 0 | |
| Callisto | 0.378 | 0.018 | 1882700 | 16.689 | 0.2 | 0.0074 | 16.689 | 0 | 0 | |
| Saturno |  |
9.449 | 95.2 | 9.54 | 29.46 | 5.51 | 0.054 | 0.43 | 62 | 26.73 |
| Titan | 0.404 | 0.023 | 1221870 | 15.945 | 0.33 | 0.0288 | 15.945 | 0 | 0 | |
| Urano |  |
4.007 | 14.6 | 19.22 | 84.01 | 6.48 | 0.047 | -0.72 | 27 | 97.77 |
| Oberon | 0.119 | 0.00051 | 583519 | 13.46 | 0.1 | 0.0014 | 13.46 | 0 | 0 | |
| Neptuno |  |
3.883 | 17.2 | 30.06 | 164.8 | 6.43 | 0.009 | 0.67 | 14 | 28.32 |
| Tritón | 0.212 | 0.00358 | 354759 | 5.877 | 157 | 0.00002 | 5.877 | 0 | 0 | |
| Plutón |  |
0.186 | 0.0022 | 39.482 | 247.9 | 17.14 | 0.25 | 6.39 | 5 | 119.59 |
| Caronte | 0.095 | 0.00025 | 17536 | 6.387 | 0.001 | 0.0022 | 6.387 | 0 | desconocido | |
| Haumea | 0.13 | 0.0007 | 43.335 | 285.4 | 28.19 | 0.19 | 0.167 | 2 | desconocido | |
| Makemake | 0.11 | desconocido | 45.792 | 309.9 | 28.96 | 0.16 | desconocido | 1 | desconocido | |
| Eris | 0.18 | 0.0028 | 67.668 | 557 | 44.19 | 0.44 | desconocido | 1 | desconocido |
Ecuaciones
B.4.1 Derivados vectoriales
Gradiente:
\ [\ nabla f (\ negrita {r}) =\ nabla f (x, y, z) =\ left (\ begin {array} {c}
\ parcial_ {x} f\\
\ parcial_ {y} f\\
\ parcial_ {z} f
\ end {array}\ derecha) =\ izquierda (\ frac {\ parcial f} {\ parcial x}\ sombrero {x} +\ frac {\ parcial f} {\ parcial y}\ sombrero {y} +\ frac {\ parcial f} {\ parcial z}\ sombrero {z}\ derecha)\]
Divergencia:
\ [\ nabla\ cdot\ negrita {v} =\ izquierda (\ parcial_ {x},\ parcial_ {y},\ parcial_ {z}\ derecha)\ cdot\ izquierda (\ begin {array} {c}
v_ {x}\\
v_ {y}\\
v_ {z}
\ end {array}\ derecha) =\ frac {v_ parcial\ _ {x}} {\ parcial x} +\ frac {\ parcial v_ {y}} {\ parcial y} +\ frac {\ parcial v_ {z}} {\ parcial z}\]
Curl:
\ [\ nabla\ veces\ negrita {A} =\ izquierda (\ parcial_ {x},\ parcial_ {y},\ parcial_ {z}\ derecha)\ veces\ left (\ begin {array} {c}
A_ {x}\
A_ {y}\\
A_ {z}
\ end {array}\ right) =\ left (\ begin {array} c}
\ parcial_ {y} A_ {z} -\ parcial_ {z} A_ {y}\\
\ partial_ {z} A_ {x} -\ parcial_ {x} A_ {z}\\
\ parcial_ {x} A_ {y} -\ parcial_ {y} A_ {x}
\ final {matriz}\ derecha)\]
B.4.2 Relatividad Especial
Transformaciones de Lorentz para las coordenadas de un fotograma S' que se mueve con una velocidad u en la dirección x positiva del fotograma S:
\ [\ begin {align}
x^ {\ prime} &=\ gamma (u)\ izquierda (x-\ frac {u} {c} c t\ derecha)\\
c t^ {\ prime} &=\ gamma (u)\ izquierda (c t-\ frac {u} {c} x\ derecha)\
\ gamma (u) &=\ frac {1} {\ sqrt {1- (u/c) ^ {2}}}
\ end {align}\]
Adición de velocidad en un sistema relativista:
\[v_{x}=\frac{u+v_{x}^{\prime}}{1+u v_{x}^{\prime} / c^{2}} \quad \text { (longitudinal) } , v_{y}=\frac{1}{\gamma(u)} \frac{v_{y}^{\prime}}{1+u v_{x}^{\prime} / c^{2}} \quad \text { (transversal) }\]