Homework #1 Notes

1. Homework #1 Notes Average: (36.1/40 (91.5%) High: 40/40 (100%) • Show all your work – hard to assign partial credit if your answer is wrong. • Several forgot to compare Betelgeuse’s radius to Sun’s radius and Earth-Sun distance. • No need to carry more than 2-3 digits throughout a problem. • Double check answers – several math errors. • No calculators for back of the envelope calculations!!

2. Homework #2 Homework #2 is due Thursday, September 20.

3. Why do all objects fall at the same rate? g = • The gravitational acceleration of an object like a rock does not depend on its mass because Mrock in the equation for acceleration cancels Mrock in the equation for gravitational force.

4. The Acceleration of Gravity (g) • Galileo showed that gis the samefor all falling objects, regardless of their mass (but most likely not from dropping objects from the Leaning Tower of Pisa). Apollo 15 demonstration David Scott, James Irwin*, Alfred Worden *no relation to me!

5. How does Newton’s law of gravity extend Kepler’s laws? • Kepler’s first two laws apply to all orbiting objects, not just planets. • Generalize acceleration due to gravity: • agravity = GM/d2 • agravity is the acceleration felt by an object when it is a distance d from a mass M • agravity is independent of the mass of the object (all objects fall to Earth with same acceleration).

6. Thought QuestionHow does the force the Earth exerts on you compare with the force you exert on it? • Earth exerts a larger force on you. • You exert a larger force on Earth. • Earth and you exert equal and opposite forces on each other.

7. Thought QuestionHow does the force the Earth exerts on you compare with the force you exert on it? • Earth exerts a larger force on you. • You exert a larger force on Earth. • Earth and you exert equal and opposite forces on each other. Newton’s 3rd Law

8. Thought QuestionWhen you jump off a diving board, how does the acceleration the Earth exerts on you compare with the acceleration you exert on it? • Earth and you exert an equal acceleration on each other. • The Earth exerts a much larger acceleration on you than you do on Earth. • You feel the acceleration from the Earth, but the Earth does not accelerate because of you.

9. Thought QuestionWhen you jump off a diving board, how does the acceleration the Earth exerts on you compare with the acceleration you exert on it? • Earth and you exert an equal acceleration on each other. • The Earth exerts a much larger acceleration on you than you do on Earth. • You feel the acceleration from the Earth, but the Earth does not accelerate because of you.

10. Four Fundamental Forces of Nature Strong and Weak force – only relevant inside the nucleus of an atom! Completely irrelevant on every day or cosmological distance scale. Electromagnetic force – most relevant in our every day life (keeps objects together), but mostly irrelevant on cosmological distance scales since + and – charges largely cancel out each other.

11. Four Fundamental Forces of Nature Gravitational force – extremely weak compared to other forces, but wins out on cosmological distance scales

12. Thought Question Gravity is always pulling you toward the center of the Earth. Why don’t you fall through the floor to the center of the Earth (or alternatively, what keeps the Earth from collapsing to a single point)?

13. Thought Question Gravity is always pulling you toward the center of the Earth. Why don’t you fall through the floor to the center of the Earth (or alternatively, what keeps the Earth from collapsing to a single point)? Answer: The electromagnetic repulsion between the atoms in your shoes and the atoms in the ground is strong enough to counteract gravity.

14. How is mass different from weight? • Mass – the amount of matter in an object (# of molecules) • Weight – the force that acts upon an object You are weightless in free-fall.

15. Thought QuestionHow would your weight change if the Earth expanded to twice its current size, but kept the same mass?

16. Thought QuestionHow would your weight change if the Earth expanded to twice its current size, but kept the same mass? g = GMEarth/Rearth2 If Rearth goes up by a factor of two, g (and your weight) go down by a factor of 4.

17. Why are astronauts weightless in space? • There is gravity in space. • Weightlessness is due to a constant state of free-fall. • Astronauts are literally “falling around” the Earth • Astronauts must conserve angular momentum

18. Center of Mass Solar System: orbiting mass is tiny compared to Sun Reality: all masses tug on all other masses Take two comparable masses, m1 and m2; they go around the center of mass. r1, r2 represent the semi-major axis of the orbits of m1, m2, respectively 

19. Center of Mass • Because of linear momentum conservation, orbiting objects orbit around their center of mass m1 * r1 = m2 * r2 Center of mass of the solar system is usually inside the Sun.

20. Newton’s Version of Kepler’s Third Law P2 = 42 * a3/(G * (m1 + m2)) or (m1 + m2) = 42 * a3 / (G * P2) P = orbital period (time for one orbit) a =average orbital distance (between centers) - aka semi-major axis of orbit (m1 + m2) = sum of object masses Here, P, m1+ m2 , and r must be in consistent units with your numerical value choice of G!

21. Let’s Derive Kepler’s Third Law Mass m1 orbits the center of mass of system at radius r1 in time P  Velocity v1 = 2 r1 / P Mass m2 orbits the center of mass of system at radius r2 in time P Velocity v2 = 2 r2 / P Force of gravity provides centripetal acceleration of masses in orbit For mass m1: m1v12/r1 = Gm1m2/(r1+r2)2 Gravity force same Centripetal force For mass m2: m2v22/r2 = Gm1m2/(r1+r2)2

22. Let’s Derive Kepler’s Third Law v12 = Gm2r1/(r1+r2)2 and v22 = Gm1r2/(r1+r2)2 Substitute for v1, v2 (2 r1 / P)2 = Gm2r1/(r1+r2)2 (2 r2 / P)2 = Gm1r2/(r1+r2)2 42r12/P2 = Gm2r1/(r1+r2)2 and 42r22/P2 = Gm1r2/(r1+r2)2 m2 = 42r1(r1+r2)2 / GP2 and m1 = 42r2(r1+r2)2 / GP2 Finally, add the two equations. 

23. Let’s Derive Kepler’s Third Law m1 + m2 = 42r1(r1+r2)2 / GP2 + 42r2(r1+r2)2 / GP2 = 42(r1+r2)3 / GP2 OR P2 = 42(r1+r2)3 / G(m1 + m2) substitute a = r1 + r2 = separation between objects = semi-major axis Formula derived for circular orbits, but also holds for elliptical orbits (algebra is more complicated).

24. Trying out Kepler’s Third Law m1 + m2 = 42(r1+r2)3 / GP2 OR P2 = 42(r1+r2)3 / G(m1 + m2) Sun-Earth m1 = Sun = 2 x 1033 g m2 = Earth = 6 x 1027 g r1 + r2 = 1 AU = 1.5 x 1013 cm G = 6.67 x 10-8 dyne cm2/g2 P = 3.16 x 107 seconds = 1 year! Earth-Moon r1 + r2 = lunar distance = 3.84 x 1010 cm P = 27.3 days (m1 + m2) = 6 x 1027 g (Earth’s mass dominates Moon’s) Units matter! Be consistent! Must convert to seconds!

25. How does Newton’s law of gravity extend Kepler’s laws? • Kepler’s first two laws apply to all orbiting objects, not just planets. • Ellipses are not the only orbital paths. Orbits can be any conic section: • bound (ellipses) • unbound • parabola • hyperbola

26. How do gravity and energy together allow us to understand orbits? • Total orbital energy (gravitational + kinetic) stays constant if there is no external force. • Orbits cannot change spontaneously. Total orbital energy stays constant and kinetic energy + grav. potential energy < 0 for a bound orbit .

27. Kinetic and gravitational energy change during orbit, but their sum is always a constant (conservation of energy)

28. Escape Velocity • If an object gains enough orbital energy, it may escape (change from a bound to unbound orbit). • Escape velocityfrom Earth ≈ 11 km/s from sea level (about 40,000 km/hr) Unbound orbits have positive total energy.

29. Escape Velocity Escape velocity defined as the velocity needed for an object to escape and come to rest (zero velocity) at an infinite distance. Efinal = KEfinal + PEfinal = ½*mvf2 – GMm/Rinf = 0 (vf = 0,Rinf = ∞) Energy must be conserved! Einitial = KEinitial + PEinitial = ½ *mv2escape – GMm/R = 0 (here, R is the radius of the gravitating object)  vescape = sqrt(2GM/R) = 11 km/s at Earth’s surface Note: escape velocity is independent of mass of orbiting object.

30. Changing an Orbit So what can make an object gain or lose orbital energy? Friction or atmospheric drag or a gravitational encounter Comet Shoemaker-Levy 9

31. How does gravity cause tides? 3 1 2 Force 1 < Force 2 < Force 3 • Moon’s gravity pulls harder on near side of Earth than on far side. • Difference in Moon’s gravitational pull stretches Earth (water more so than land). • Note tidal bulges on BOTH sides of the Earth. • Sun also causes tides on Earth (to a lesser extent)

32. Tides and Phases Size of tides depends on phase of Moon. New/Full Moon: Moon and Sun work together  spring tide 1st/3rd Quarter Moon: Moon and Sun somewhat offset each other (but Moon wins out)  neap tides

33. Tides and Phases Size of tides depends on phase of Moon. New/Full Moon: Moon and Sun work together  spring tide 1st/3rd Quarter Moon: Moon and Sun somewhat offset each other (but Moon wins out)  neap tides

34. Tidal Force • The difference in gravity on the near side and the center of a planet/moon is called the tidal forceon the object. For an object of radius r being pulled on by an object of mass M at a distance d: acenter = GM/d2 anear=GM/(d-r)2 • atidal = anear – acenter = GM/(d-r)2 – GM/d2 = GM/d2 [1/(1-r/d)2 - 1] For small r/d, 1/(1-r/d)2 ~ 1+2r/d (Taylor expansion)  atidal = 2GMr/d3 tidal acceleration VERY dependent on distance, d

35. Consequently, the bulge leads the actual location of the Moon.

37. Tidal Friction • Raising a tidal bulge dissipates energy, which can cause heating. The energy typically comes out of the rotation, so the object is slowed down. • Tidal friction gradually slows Earth’s rotation (and makes the Moon get farther from Earth – 1.5 inches/year). • The Moon once orbited faster; tidal friction caused it to “lock” in synchronous rotation  same side always faces Earth.

38. Tidal Friction • As Moon’s orbit gains angular momentum, Earth rotation slows (17 microseconds/year) – sidereal day was 15 minutes shorter 65 million years ago. • This would lead (in 50 billion years) to the Moon’s orbit, Moon’s spin, and Earth’s spin all being 47 days long. • However, in ~2 billion years, the expected increase in solar output will vaporize the Earth’s oceans  greatly reduced tides