Newton s Laws of Motion and Gravity ASTR 2110 Sarazin. Space Shuttle

Transcription

1 Newton s Laws of Motion and Gravity ASTR 2110 Sarazin Space Shuttle

2 Discussion Session This Week Friday, September 8, 3-4 pm Shorter Discussion Session (end 3:40), followed by: Intro to Astronomy Department for Potential Astronomy or Astronomy-Physics Majors (if not interested in majors, no need to stay)

3 Welcome Party Friday, September 8, 4:30-6 pm Food and drinks. Meet the Astronomy Department and NRAO McCormick Observatory Will walk interested students over to Department after Discussion Session. Walk or rides provided up O Hill

4 Isaac Newton ( )

5 Newton s 1 st Law F = 0 v = constant a = 0 Unless subject to a force, an object will continue to move at a constant velocity Objects at rest stay at rest Objects in motion maintain same speed and direction

6 F = 0 a = 0 F a? Newton s 2 nd Law F = ma = m d v dt = d dt p m mass (total amount of matter in object) p m v momentum a = F m equation of motion

7 Newton s 3 rd Law Is physics just too tough? N atoms ~ N forces = N atoms (N atoms 1)/2 ~ 10 48!! F = F F12 F 21 Every force produces an equal and opposite force

8 Momentum: d p i dt p tot = = j Conservation Laws Closed system F ij i p i d p tot = F ij = 1 " dt i, j 2 % $ F ij + F ji ' = 1 # i, j j,i & 2 i, j p tot = constant, momentum is conserved ( F ij + F ) ji = 0

9 Center of Mass p tot = v CM r CM i i i m i vi m i vi m i ri = constant i m i = constant m i moves at constant velocity i

10 Conservation Laws Energy: Object of mass m moving in a force field m d v dt = F Now, do dot product with v mv d v dt = v F = F d r dt v d v dt = 1 d v 2 = 1 2 dt 2 2 v d v dt 1 2 m d v2 = F d r Now, integrate both sides dt dt 1 2 mv2 = F d r + constant dt

11 Conservation Laws Energy: 1 2 mv2 = Define: F d r + constant KE 1 2 mv2 kinetic energy PE F d r potential energy E = KE + PE = constant Total energy is conserved

12 Conservation Laws Angular Momentum: L r p angular momentum Assume: F ij r ij Central force F 12 r 12 F21 d L tot dt = i d dt ( r i p ) i

13 Conservation Laws Angular Momentum: r 12 d L tot dt = = = i i i, j d dt ( r i p ) i v i m i vi + r i F i ( ) r i F i, j F 12 F21 = 1 2 i, j r i F i, j + r j F j,i ( )

14 Conservation Laws Angular Momentum: r 12 d L tot dt = 1 2 i, j r i F i, j + r j F j,i ( ) F 12 F21 but F j,i = F i, j d L tot = 1 ( r i r ) j F i, j = 1 dt 2 i, j 2 F ij r ij Central force i, j r ij F i, j d L tot dt = 0, L tot = constant

15 Conservation Laws Something for nothing? Conservation Laws Momentum Energy Angular Momentum çè Symmetries of nature çè All places are the same Space is homogeneous çè All times are the same çè All directions are the same Space is isotropic

16 Turning now to gravity

17 No force move in straight line Planet orbits curved (ellipses) must be a force acting Prob. 2.1 in homework a cent = v2 r Force due to Sun What is force? Planetary Motions êr Force toward Sun Sun force planet

18 Lunar Motion Moon s orbit curved must be a force Force due to Earth: What is force?

19 Lunar Motion Moon force force

20 Gravity Gravity is force in motion of Moon Gravity is force in planetary motion

21 Galileo and Gravity Galileo also studied gravity on Earth Pre-Galileo: Heavy objects fall faster Objects fall at a constant speed Galileo: Objects fall at a constant acceleration All objects fall with the same acceleration (no air friction)

22 Properties of Gravity Newton s 2 nd Law: a = F/m = g = constant Can only be true for everything if F m Force on object proportional to its mass Newton s 3 rd Law: F 21 m 2 force on 2 due to 1 F 12 = - F 21 F 12 m 2 force on 1 due to 2 m 1 m F F Force due to object proportional to its mass

23 Properties of Gravity Force is proportional both to mass of source and mass of object being affected F 12 m 1 F 12 m 2 F 12 m 1 x m 2

24 Properties of Gravity Compare acceleration of Moon, apple Acceleration of Moon much smaller. Distance to Moon much greater. Assume F = m 1 m 2 f(d) decreasing function of distance d

25 Properties of Gravity Assume F = m 1 m 2 f(d) decreasing func. d Moon = 3.84 x 10 5 km, d Earth = 6378 km v Moon = 2πd Moon /P Moon, P moon =27.3 d a Moon = v Moon2 /d Moon =0.272 cm/s 2 a Earth = g = 981 cm/s 2 = 3600 a Moon =60 2 a Moon d Moon = 60 d Earth F α d -2

26 Newton s Law of Gravity F 12 = - G m 1 m 2 / d 2 d d = distance m 1 m F F G = constant of nature = 6.67 x 10-8 cm 3 / gm / s 2 = 6.67 x m 2 / kg / s 2

27 Newton s Law of Gravity F 12 = G m 1m 2 r 12 2 PE = G m 1m 2 r 12 ê r d m 1 m F F m 1 a1 = G m 1 m 2 r 12 2 ê r

28 Newton s Law of Gravity m 1 a1 = G m 1 m 2 r 12 2 ê r d m 1 m F F Inertia Three Roles for Mass ma Gravity Source m 2 active Gravity Effect m 1 passive Why mass? Inertia and passive gravity terms cancel why?

29 Spherical Object F = G M(r)m r 2 ê r M(r) mass interior to r r m

30 One Body Solvable N-Body (N 3) Two Body N-Body Problem Fixed mass Generally, no analytic solution. Solve numerically on a computer Six Equations, but total momentum conserved è Only 3 independent equations, same as 1-body 2-Body Problem = 1-Body Problem in CM frame, solvable. m

31 2-Body Problem v r v 1 r 1 m 1 CM r 2 m 2 v 2 = r μ r = r 2 r 1 r 1 = v = v 2 v 1 v 1 = m 2 m 1 + m 2 r r 2 = m 2 m 1 + m 2 v v 2 = m 1 m 1 + m 2 r m 1 m 1 + m 2 v µ m 1m 2 m 1 + m 2 "reduced mass"

32 2-Body Problem v r v 1 r 1 m 1 CM r 2 m 2 v 2 = r μ KE = 1 2 µv2 PE = G m 1 m 2 r E = KE + PE = constant L = µ r v = constant

33 Newton Derived Kepler s Law Derived, generalized, corrected Kepler s laws Treat Solar System as a series of 2-body problems (Sun and each planet), since most of gravity is from the Sun

34 Kepler s 2 nd Law Equal areas in equal times d A = 1 2 r perpendicular component of d s = 1 2 r d s = 1 2 r v dt r d A = 1 2m = 1 2m da dt = constant [ r (mv) ]dt = 1 2m L dt = constant dt r p ( )dt True for any central force (not just gravity) ds = v dt

35 Kepler s 3 nd Law P 2 α a 3 Do for a circular orbit m a = F (1-body) µ a = F (2-body) m 1 m 2 m 1 + m 2 " $ # v2 r v 2 = G(m 1 + m 2 ) r v = 2πr P P 2 =! # " % ' e r = G m 1m 2 & r 2 2πr P 4π 2 a 3 G(m 1 + m 2 ) $ & % 2 Not quite Kepler, depends on mass! er v = G(m 1 + m 2 ) r = v 2 4π 2 r 2 = G(m 1 + m 2 ) P 2 r

36 Kepler s 1 st Law Orbits ellipses, Sun at focus Use conservation of L and E L planar orbit, L ( ) r = L2 G m 1 m 2 µ 1+ ecosθ e = 1+ 2EL 2 G 2 m 1 2 m 22 µ 1 cosθ +1 e <1 1+ ecosθ > 0 r remains finite (bound) e 1 1+ ecosθ 0 r (unbound) e <1 E < 0 Conic section

37 Kepler s 1 st Law Energy Eccentricity Shape Bound? E < 0 e < 1 Ellipse (circle) Bound (can t get apart) E = 0 e = 1 Parabola Just unbound E > 0 e > 1 Hyperbola Unbound Why does energy determine orbit? Conserved Unbound è r è PE è 0 E = KE = ½ m v 2 0

38 ( r = L2 G m 1 m 2 µ ) 1+ ecosθ r = ( ) a 1 e2 1+ ecosθ Kepler s 1 st Law Compare to previous equation for ellipse Only for ellipse E = 1 2 G m 1 m 2 a = constant (bound orbit) PE = G m 1 m 2 r Odd: E = 1 2 PE??

39 Consider A Virial Theorem No static gravitational equilibrium since force always attractive KE > 0 da dt = i m i v i r i (Why? Trust me.) i m ia i r ( i + m ivi v i ) " = m iv 2 i + F i r % $ ' # & i = 2KE + F i r i

40 Virial Theorm Now, consider average of da/dt over a long time da dt = A(t) t So either A(t) A(0) t da dt da dt t + A(0) as t mean value thm of calculus 0 or A(t) A v i r i so either v or r not bound! So, da dt 0 over a long time

41 Virial Theorm da dt 2 KE = = 2 KE + = + = i i, j pairs F i r i i F i r i G m i m j ri r j r i r 3 j ( ) G m i m j ri r j r i r 3 j ( ) 0 over a long time r i Do by pairs i,j r i + G m m j i r j r i r j r 3 i ( ) r j

42 Virial Theorm 2 KE = pairs G m i m j r i r 3 j " #( r i r j ) r i r i r j ( ) r j $ % = G m i m j r i r ( r 3 i r ) 2 j = pairs j pairs G m i m j r i r j 2 KE = PE KE = 1 2 PE E = KE + PE = constant E = E E = 1 2 PE = KE

43 Electromagnetism and Light ASTR 2110 Sarazin Laser Guide Star at Telescope

44 Four Forces in Nature Strong (Nuclear) Force: ~10 1 Electromagnetic Force: ~10-2 α = e 2 /ħc 1/137 Weak (Nuclear) Force: ~10-18 Gravity: ~10-40 But gravity applies to everything, always attractive

45 Electromagnetic Forces E electric field B magnetic field # F = q E v B & % + ( $ c ' q charge F 12 = q q 1 2 ê r 2 r Coulomb s Law

46 Maxwell s Equations Complete theory of electricity and magnetism Electric charges make electric field Coulomb s Law or Gauss s Law

47 Static Electricity

48 Maxwell s Equations Complete theory of electricity and magnetism Electric charges make electric field charges E

49 Maxwell s Equations Complete theory of electricity and magnetism Electric charges make electric field No magnetic charges (magnetic monopoles) charges E no magnetic charges B

50 Maxwell s Equations Complete theory of electricity and magnetism Electric charges make electric field No magnetic charges (magnetic monopoles) Moving charges (currents) make magnetic fields Ampere s Law

51 Electromagnets

52 Maxwell s Equations Complete theory of electricity and magnetism Electric charges make electric field No magnetic charges (magnetic monopoles) Moving charges (currents) make magnetic fields charges E no magnetic charges currents B

53 Maxwell s Equations Complete theory of electricity and magnetism Electric charges make electric field No magnetic charges (no magnetic monopoles) Moving charges (currents) make magnetic fields Changing magnet fields make electric fields Faraday s Law

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