16. Some equations and constants#

16.1. Physical constants#

Table 16.1 Physical constants#

Name

Symbol

Value

Speed of light

\(c\)

\(3.00 \cdot 10^{8}\;\mathrm{m}/\mathrm{s}\)

Elementary charge

\(e\)

\(1.60 \cdot 10^{-19}\;\mathrm{C}\)

Electron mass

\(m_e\)

\(9.11 \cdot 10^{-31}\;\mathrm{kg} = 0.511\;\mathrm{MeV}/c^2\)

Proton mass

\(m_p\)

\(1.67 \cdot 10^{-27}\;\mathrm{kg} = 938\;\mathrm{MeV}/c^2\)

Gravitational constant

\(G\)

\(6.67 \cdot 10^{-11}\;\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2\)

Gravitational acceleration

\(g\)

\(9.81\;\mathrm{m}/\mathrm{s}^2\)

Boltzmann’s constant

\(\kB\)

\(1.38 \cdot 10^{-23}\;\mathrm{J}/\mathrm{K}\)

Gas constant

\(R\)

\(8.31 \; \mathrm{J}/\mathrm{K} \cdot \mathrm{mol}\)

Stefan-Boltzmann constant

\(\sigma\)

\(5.67 \cdot 10^{-8} \; \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}^4\)

Avogadro’s number

\(N_\mathrm{A}\)

\(6.02 \cdot 10^{23} \; \mathrm{mol}^{-1}\)

Planck’s constant

\(h\)

\(6.63 \cdot 10^{-34}\;\mathrm{J}\cdot\mathrm{s}\)

\(\hbar = h / 2 \pi\)

\(1.05 \cdot 10^{-34}\;\mathrm{J}\cdot\mathrm{s}\)

16.2. Moments of inertia#

Table 16.2 Moments of inertia, all about axes of symmetry through the center of mass.#

Object

Moment of inertia

Thin stick (length \(L\))

\(\frac{1}{12} M L^2\)

Ring or hollow cylinder (radius \(R\))

\(M R^2\)

Disk or solid cylinder (radius \(R\))

\(\frac12 M R^2\)

Hollow sphere (radius \(R\))

\(\frac23 M R^2\)

Solid sphere (radius \(R\))

\(\frac25 M R^2\)

Rectangle (size \(a \times b\)), perpendicular axis

\(\frac{1}{12} M (a^2 + b^2)\)

Rectangle (size \(a \times b\)), axis parallel to side \(b\)

\(\frac{1}{12} M a^2\)

16.3. Solar system objects#

Table 16.3 Characteristics of the Sun, Earth and Moon.#

Sun

Earth

Moon

Mass (kg)

\(1.99 \cdot 10^{30}\)

\(5.97 \cdot 10^{24}\)

\(7.35 \cdot 10^{22}\)

Mean radius (m)

\(6.96 \cdot 10^8\)

\(6.37 \cdot 10^6\)

\(1.74 \cdot 10^6\)

Orbital period (s)

\(6 \cdot 10^{15}\)

\(3.16 \cdot 10^7\)

\(2.36 \cdot 10^6\)

(200 My)

(365.25 days)

(27.3 days)

Mean orbital radius (m)

\(2.6 \cdot 10^{20}\)

\(1.50 \cdot 10^{11}\)

\(3.85 \cdot 10^{8}\)

Mean density (kg/m\(^3\))

\(1.4 \cdot 10^3\)

\(5.5 \cdot 10^3\)

\(3.3 \cdot 10^3\)
Table 16.4 Properties of a number of solar system objects.#

Name

Symbol

Equatorial radius

Mass

Mean orbit radius

Orbital period

Inclination

Orbital eccentricity

Rotation period

Confirmed moons

Axial tilt

Mercury

☿

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

Earth

♁

1

1

1

1

7.25

0.017

1

1

23.44

Moon

☾

0.272

0.0123

384399

27.32158

18.29-28.58

0.0549

27.32158

0

6.68

Mars

♂

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

Jupiter

♃

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

Ganymede

0.413

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

Saturn

♄

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

Uranus

⛢

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

Neptune

♆

3.883

17.2

30.06

164.8

6.43

0.009

0.67

14

28.32

Triton

0.212

0.00358

354759

5.877

157

0.00002

5.877

0

0

Pluto

♇

0.186

0.0022

39.482

247.9

17.14

0.25

6.39

5

119.59

Charon

0.095

0.00025

17536

6.387

0.001

0.0022

6.387

0

unknown

Haumea

0.13

0.0007

43.335

285.4

28.19

0.19

0.167

2

unknown

Makemake

0.11

unknown

45.792

309.9

28.96

0.16

unknown

1

unknown

Eris

0.18

0.0028

67.668

557

44.19

0.44

unknown

1

unknown

Properties of a number of solar system objects. Equatorial radii and masses are compared to those of Earth (see Table 16.3). Orbital properties are around primary (the sun for (dwarf) planets, the planet for moons). Orbital radii and periods for planets again compared to Earth, for moons in kilograms and days. Rotation period for all objects in days. Inclination and axial tilt in degrees. Data from NASA planetary fact sheets [1].

16.4. Material properties#

Table 16.5 Properties of some common solid materials.#

material

\(\rho\) (\(\mathrm{kg}/\mathrm{m}^3\))

\(E\) (\(\mathrm{GPa}\))

\(B\) (\(\mathrm{GPa}\))

\(G\) (\(\mathrm{GPa}\))

\(c\) (\(\mathrm{J}/\mathrm{kg}\cdot\mathrm{K}\))

\(k\) (\(\mathrm{W}/\mathrm{m}\cdot\mathrm{K}\))

aluminium

2700

70

75

25

900

237

bone

16

8

80

concrete

2400

20

copper

8940

110

140

44

386

401

glass

60

50

25

753

0.8

granite

45

45

20

hardwood

700

15

10

iron

7870

210

160

77

447

80.4

marble

60

70

20

steel

200

160

75

ice (\(-10^\circ\mathrm{C}\))

916.7

2050

pine wood

373

7.5

5

1400

0.11

Properties of some common solid materials, measured at room temperature and pressure, unless indicated otherwise: densities \(\rho\), Young’s \(E\), bulk \(B\) and shear \(G\) moduli, specific heats per unit mass \(c\) and thermal conductivities per unit length \(k\). The ‘\(\mathrm{G}\)’ in \(\mathrm{GPa}\) stands for ‘giga’, or \(10^9\).

Table 16.6 Properties of some common liquid materials#

material

\(\rho\) (\(\mathrm{kg}/\mathrm{m}^3\))

\(\eta\) (\(\mathrm{Pa}\cdot\mathrm{s}\))

\(T_\mathrm{m}\) (\(\mathrm{K}\))

\(L_\mathrm{f}\) (\(\mathrm{kJ}/\mathrm{kg}\))

\(T_\mathrm{b}\) (\(\mathrm{K}\))

\(L_\mathrm{v}\) (\(\mathrm{kJ}/\mathrm{kg}\))

\(c\) (\(\mathrm{J}/\mathrm{kg}\cdot\mathrm{K}\))

ethanol

789

\(1.07 \cdot 10^{-3}\)

159

109

351

879

2.43

glycerol

1260

1.2

291

201

563

1007

2437

mercury

13530

\(1.53 \cdot 10^{-3}\)

234

11.3

630

296

140

water

1000

\(8.94 \cdot 10^{-4}\)

273

334

373

2257

4184

Properties of some common liquid materials: densities \(\rho\), viscosities \(\eta\), melting and boiling points \(T_\mathrm{m}\) and \(T_\mathrm{b}\), heats of fusion \(L_\mathrm{f}\) and vaporization \(L_\mathrm{v}\) and specific heats per unit mass \(c\). Densities, viscosities and specific heats measured at room temperature and pressure.

Table 16.7 Melting and boiling points of some common materials, with their respective heats of transformation, measured at atmospheric pressure.#

Material

melting point (\(K\))

\(L_\mathrm{f}\;(\mathrm{kJ}/\mathrm{kg})\)

boiling point (\(K\))

\(L_\mathrm{v}\;(\mathrm{kJ}/\mathrm{kg})\)

alcohol

159

109

351

879

copper

1357

205

2840

4726

lead

601

24.7

2013

858

mercury

234

11.3

630

296

water

273

334

373

2257

16.5. Equations#

16.5.1. Vector derivatives#

Gradient:

(16.1)#\[\begin{split} \bm{\nabla} f(\bm{r}) = \bm{\nabla} f(x, y, z) = \left( \begin{array}{c} \partial_x f \\ \partial_y f \\ \partial_z f \end{array}\right) = \left( \frac{\partial f}{\partial x} \hat{x} + \frac{\partial f}{\partial y} \hat{y} + \frac{\partial f}{\partial z} \hat{z} \right). \end{split}\]

Divergence:

(16.2)#\[\begin{split} \bm{\nabla} \cdot \bm{v} = \left( \partial_x, \partial_y, \partial_z \right) \cdot \left( \begin{array}{c} v_x \\ v_y \\ v_z \end{array} \right) = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}. \end{split}\]

Curl:

(16.3)#\[\begin{split} \bm{\nabla} \times \bm{A} = \left( \partial_x, \partial_y, \partial_z \right) \times \left( \begin{array}{c} A_x \\ A_y \\ A_z \end{array} \right) = \left( \begin{array}{c} \partial_y A_z - \partial_z A_y \\ \partial_z A_x - \partial_x A_z \\ \partial_x A_y - \partial_y A_x \end{array} \right). \end{split}\]