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L04:Complex Numbers Spring 2020 Introduction to General Physics I Maurice H.P.M. van Putten Department of Physics and Astronomy MO and WE Online / March 2020 (c)2020 van Putten 1 Week 4 Mater in moton Johannes Kepler (1571-1630) Konstantin E. Tsiolkovsky 1857-1935 (c)2020 van Putten 2 Contents - Week 4 4.1 Rocket flight: Tsiolkovsky equation 4.2 Vectors in orbital motion 4.3 Atoms (c)2020 van Putten 3 NASA Artemis Mission https://www.nasa.gov/specials/artemis/ (c)2020 van Putten 4 NASA Artemis Mission https://www.nasa.gov/specials/artemis/ (c)2020 van Putten 5 .. On to Mars https://www.nasa.gov/topics/moon-to-mars (c)2020 van Putten 6 Radiation exposure (RAD=radiation assessment detector) Richard A. Kerr Science 2013;340:1031 Copyright © 2013, American Association for the Advancement of Science Challenges 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 (c)2020 van Putten 8 Rocket engines SpaceX RS-68 (NASA) https://www.nasa.gov/multimedia/imagegallery/iotd.html (c)2020 van Putten 9 Propulsion momentum momentum outflow in gas gain in rocket initial mass m(t) v(t) final mass T t T t (c)2020 van Putten 10 Newton’s 3rd law (c)2020 van Putten 11 Newton’s 3rd law From discrete to continuous picture Δm m − Δm 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 (c)2020 van Putten 12 Tsiolkovsky equation (I) Now integrate... ⎛ ⎞ dm / dt dvr / dt mi − = : vr = vg ln⎜ ⎟ m vg ⎝ m f ⎠ 5 m(t) = mi − m!t 4.5 4 ⎛ mi ⎞ 3.5 vr (t) = vg ln⎜ ⎟ velocity 3 ⎝ mi − m!t ⎠ 2.5 (a.u.) Tsiolkovsky / ideal rocket equation 2 William Moore 1813 1.5 1 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.) (c)2020 van Putten 13 Include lift-off (c)2020 van Putten 14 Tsiolkovsky equation (II) 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 (c)2020 van Putten 15 Summary Key points of classical mechanics We are witnessing a renaissance in human exploration of space, now focused on establishing a permanent presence on the Moon and Mars 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. (c)2020 van Putten 16 Contents - Week 4 4.1 Rocket flight: Tsiolkovsky equation 4.2 Vectors in orbital motion 4.3 Atoms (c)2020 van Putten 17 Planetary motion: vectors Planetary motion Newton’s theory of gravitation (c)2020 van Putten 18 Circular motion Vectors in circular motion: Notation: Position vector has constant length, changes only in direction Velocity vector has constant length, changes only in direction (c)2020 van Putten 19 Position vectors Length of position vector Unit vector Unit vector has length 1 (c)2020 van Putten 20 Acceleration and force vectors By Newton’s theory of gravitation Work performed is zero (c)2020 van Putten 21 Polar coordinates H in polar coordinates Polar coordinate system rotates along with the moon 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 (c)2020 van Putten 22 Circular motion Special case: circular motion Constant distance Earth-Moon: r! = 0 Total energy is a constant of motion 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). (c)2020 van Putten 23 Circular motion Circular motion 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 24 Kepler’s observational test (c)2020 van Putten 25 Orbital angular momentum Orbital angular momentum (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 (c)2020 van Putten 27 Total energy once more Total energy - once more 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 (c)2020 van Putten 28 Equations of motion Equations of motion 1 H = mj 2 (u'2 + u2 ) − GMmu 2 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 (c)2020 van Putten 29 Angular momentum: Kepler and Newton Angular momentum circles ellipses 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 (c)2020 van Putten 30 Summary Copernicus heliocentric solar system (c)2020 van Putten 31 Contents - Week 4 4.1 Rocket flight: Tsiolkovsky equation 4.2 Vectors in orbital motion 4.3 Atoms (c)2020 van Putten 32 Einstein’s legacy: atoms, spacetime, energy (c)2020 van Putten 33 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 (c)2020 van Putten (c)2015 van Putten 34 Electron orbitals (c)2020 van Putten (c)2015 van Putten 35 Building atoms (c)2020 van Putten (c)2015 van Putten 36 STM of gold (c)2020 van Putten (c)2015 van Putten 37 A boy and his atom (IBM 2012) (c)2015 van Putten (c)2020 van Putten 38 Matter is build up of atoms 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. (c)2020 van Putten (c)2015 van Putten 39 Atoms in motion Atoms in motion 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 ⊥ (c)2020 van Putten (c)2015 van Putten 40 Ideal gas Energy and pressure of an ideal gas “Gas in a box” Feynman Lectures of Physics Vol I Ch1 (www.feymanlectures.caltech.edu/I_01.html) (c)2020 van Putten (c)2015 van Putten 41 Avogadro’s number Avogadro’s number 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. (c)2020 van Putten (c)2015 van Putten 42 Kinetic energy and temperature Translational thermal energy Translational kinetic energy: 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).
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