L04:Complex Numbers

Home , Collision

Spring 2020

Introduction to General Physics I

Maurice H.P.M. van Putten

Department of Physics and Astronomy

MO and WE

Online / March 2020

Johannes Kepler (1571-1630) Konstantin E. Tsiolkovsky 1857-1935

4.1 Rocket ﬂight: Tsiolkovsky equation

4.2 Vectors in orbital motion

4.3 Atoms

https://www.nasa.gov/specials/artemis/

https://www.nasa.gov/topics/moon-to-mars

Richard A. Kerr Science 2013;340:1031

Safety: protection against cosmic radiation during travel and stay on Mars

Sustainability: self-sustainable food supply “home- grown” on Mars, essential nutrition

Well-being: health, social and entertainment, coffee (?)

Reduced gravity: zero during travel, 1/3 on Mars

Contact: no real-time contact with Earth (delays on the order of 15 min or more)

.. https://en.wikipedia.org/wiki/Human_mission_to_Mars

Technical: new rocket designs

SpaceX RS-68 (NASA)

https://www.nasa.gov/multimedia/imagegallery/iotd.html

momentum momentum outﬂow in gas gain in rocket

initial mass m(t) v(t) ﬁnal mass

Δpg Δpr

0 = Δpg + Δpr = −Δm vg + (m − Δm)Δvr

Δm Δvr ≅ (ΔmΔvr is second order) m vg dm Δm = − Δt (m=m(t) is mass of the rocket) dt dm / dt dv / dt − = r m vg

5 m(t) = mi − m!t 4.5

4 ⎛ mi ⎞ 3.5 vr (t) = vg ln⎜ ⎟ velocity 3 m − m!t ⎝ i ⎠ 2.5 (a.u.) Tsiolkovsky / ideal rocket equation 2 William Moore 1813 1.5

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (a.u.)

m m i i vr,f ≃ vg ln + g · ( mi − mf ) m

To ﬁnalize, generalize to include GMm Gm U (z) = − , g(z) = − N R + z (R + z)2

In reality, the greatest challenges are perhaps non- technical, concerning radiation safety, self-sustainable food supplies and well-being of crew over the course of multi- year visits.

Key technology is the development of new rockets ensuring safety over long-duration travel (months to Mars), large cargo and re-usable.

4.1 Rocket ﬂight: Tsiolkovsky equation

4.2 Vectors in orbital motion

4.3 Atoms

Newton’s theory of gravitation

Vectors in circular motion:

Notation:

Position vector has constant length, changes only in direction

Velocity vector has constant length, changes only in direction

Length of position vector

Unit vector

Unit vector has length 1

By Newton’s theory of gravitation

Work performed is zero

d Rotation of an ONB {i', j '} = {i ,i }, i = rˆ, i = ϕ!i r ϕ r dt r ϕ ϕ {i,j} {i’,j’} r! = r!ir + (rϕ!)iϕ y-axis 2 2 2 v = r! = r! + r ϕ! unit circle iϕ A i r ϕ x-axis 1 1 1 GMm H = mr!2 + mr2ϕ! 2 − 2 2 r

total kinetic energy potential energy

1 GMm H = mr2ϕ! 2 − 2 r d H = 0 : mr2ϕ!ϕ!! = 0 ⇒ϕ!! = 0 dt

whereby the angular velocity ω = ϕ!

is constant in time. The moon circles the Earth about once per month - forever.

Newton’s gravitational force is centripetal (zero torque).

1 GMm Hamiltonian H = mr2ω 2 − < 0 Bound state! 2 r

v2 GM GM = , v = ω R ⇒ ω 2 = R R2 R3

2π ω = ⇒ P2 ∝ R3 P

Johannes Kepler (1571-1630)

(c)2020 van Putten 26 Conservation of angular momentum Angular momentum conservation Speciﬁc angular momentum j = r × v d j = r × a + v × v = r × a dt GMm Newton’s centripetal acceleration a = − rˆ r2 d j = r × a = 0 ⇒ j = const. dt unit normal to iz the orbital plane j = r2ϕ!i z j = j = r2ϕ! orbital plane

2 1 2 1 2 2 GMm 1 2 mj GMm H = mr! + mr ϕ! − = mr! + 2 − 2 2 r 2 2r r 1 Mobius transformation u = u(ϕ), u = , r 1 r! = − u'ϕ!, ! 2 ju2 2 ϕ = u

1 H = mj 2 (u'2 + u2 ) − GMmu 2 kinetic energy potential energy

GM 0 = H! = H 'ϕ! ⇒ H ' = 0 u''+ u = j 2

2 GM 1 j / GM ⎛ j 2 A ⎞ u = + Acosϕ ⇒ r = = e = j 2 u 1+ ecosϕ ⎝⎜ GM ⎠⎟

Newton’s theory (conserving energy and angular momentum) gives Kepler’s elliptical orbits

1 ΔA = r2Δϕ 2

dA π R2 1 2π 1 1 = = × R2 = R2ϕ! = j dt P 2 P 2 2

Kepler: equal areas traced out in equal times Newton: conservation of angular momentum

Newton’s force is purely centripetal, exercising zero torque

Copernicus heliocentric solar system

4.1 Rocket ﬂight: Tsiolkovsky equation

4.2 Vectors in orbital motion

4.3 Atoms

Electron shells 1 Angstrom in radius

Nucleus (protons and neutrons) Nucleus is ~ 1/100,000 smaller

Atomic mass:

# protons (1.67e-24 g) # neutrons (1.67e-24 g) ------+

Atomic number:

#protons

van Gogh

Einstein’sAtoms treatment - andof Brownian their motion: structure matter is made within up of atoms By modern STM, we can now image atoms directly

Atoms and elementary particles:

Atoms are the basic building blocks of the matter than we see, feel and taste.

The only “classical” element about it, is that we can count atoms.

Atoms are tiny: ~ 1 Angstrom (1e-10 meters), ~ 1e24 in 1 gram of sugar!

Nature of atoms is “quantum physics” in constitution and radiation properties.

pf = − pi Momentum: p = mv 1 Kinetic energy: E = mv2 k 2 p i Δp Force: F = Δt

“Bounce against a wall”: reversal of momentum orthogonal to the wall

Bouncing rate: ν [1/s] or [Hz] (“Hertz”)

Δp Average force on the wall: = v < p > Δt ⊥

“Gas in a box”

Feynman Lectures of Physics Vol I Ch1 (www.feymanlectures.caltech.edu/I_01.html)

Avogadro’s EVEN hypothesis [1811]:

At the same temperature and pressure, Equal Volumes of gas contain Equal Numbers of particles (atoms or molecules, what the case may be).

Nobel Prize winner Linux Pauling [1956, Science, 124, 708]: Avogadro's work "forms the basis of the whole of theoretical chemistry" and is "one of the greatest contributions to chemistry that has ever been made.”

The molar volume of all ideal gases at 0° C and a pressure of 1 atm. is 22.4 liters Avogadro's number: number of atoms in 12 grams C-12 (“one mole of carbon 12”)

Current usage: “1 mole" refers to Avogadro’s number of whatever, e.g., 1 mole of atoms, ions, radicals, electrons, quanta or sand on the beach.

1 1 mv2 = m (u2 + v2 + w2) 2 2 Using a Cartesian coordinate system (x, y, z) with unit vectors (i, j, k) along the x − ,y− and z−axis and component velocities (u, v, w).

k v = u i + v j + w k z-axis j i v w Mean (translational) kinetic energy: u v y-axis 1 1 x-axis mv2 ≡ k T ⟨ 2 ⟩ 2 B

For an ideal gas, pressureExercise scales linearly (scaling) with temperature.

N particles in a cube of linear size L and volume L3 with typical velocity V. The typical force by particle collisions on walls scales linearly with collision frequency and particle momentum. After all, each collision is a reﬂection, changing momenta by a change in sign of the normal component of momentum: . Recall that Δp⊥ = − 2p⊥ Particle momenta p = mV scale linearly with V. Collision frequency ν scales linearly with V/L.

The force imparted by one particle scales with

2 . νΔp⊥ ∝ νp ∝ mV /L = 2Ek /L

The total force on the walls is 2. Deﬁned by force per unit area of the walls, F ∝ EkL pressure satisﬁes 2 . Taking mean values, we arrive at p = F/L = Ek p ∝ T.

Pressure and temperature The result is commonly expressed in terms of gas temperature and pressure:

½ kBT = translational kinetic energy per particle per direction

p = force on the wall per unit surface area

Note: kB = R / NA = 1.38x10-16 erg K-1

NA = 6x1023 = mol)

erg = 10-7 Joule

[erg] = g cm2/ s2, [J] = kg m2/ s2

AtomsKepler’s ratio P2/R3 Atoms consist of a nucleus with one or more protons Nuclei are surrounding by (much larger) electron orbits

Ideal gas theory Described by pressure, temperature and the number of particles per unit volume, independent of the type of atom (or molecule).