Ch. 8 Universal Gravitation Part 1: Kepler`s Laws
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Ch. 8 Universal Gravitation Objectives: Section 8.1 Motion in the Heavens and on Earth Part 1: Kepler’s Laws Johannes Kepler Johannes Kepler (1571-1630) was an assistant to the Danish astronomer Tycho Brahe. Kepler was convinced that geometry and mathematics could be used to explain the motion of the planets. Unlike Brahe, he used a heliocentric model. Objectives Relate Kepler’s laws of planetary motion to Newton’s law of universal gravitation. Calculate the periods and speeds of orbiting objects. Describe the method Cavendish used to measure G and the results of knowing G. Tycho Brahe I spent 26 years making star charts using this quadrant and no telescope. Kepler’s Laws Johannes Kepler discovered 3 laws from the motion of the planets as mapped by Tycho Brahe 1. 2. 3. I thought the sun went around the Earth and the planets went around the Sun. Planetary orbits are elliptical not circular Orbits sweep out equal areas in equal times T2/r3 = k where k = “constant” constant” I used Tycho Brahe’s data to come up with my laws. It took him 20 years to collect it. 1 Kepler’s 1st Law Kepler’s 1st Law 1. The orbits of the planets are ellipses, with the sun at one focus. (Law of Ellipses) Orbits are ellipses The Sun is at one of the foci The closer the planets are to one another, the more circular the orbit. Kepler’s 2nd Law Kepler’s 2nd Law 2. 2. An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (Law of Equal Areas) Deals with speed… faster when closer to the sun. Kepler’s 3rd Law 3. The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (Law of Harmonies) Orbits sweep out equal areas in equal times The Sun is at one of the foci Kepler’s 3rd Law T2/r3 = k where k = “constant” constant” T = the period for 1 revolution r = the average radius of the elliptical orbit So for every orbiting body everywhere, this ratio is true (TA/TB)2 = (rA/rB)3 2 Universal Gravitation Part 2: Universal Gravitation Isaac Newton 24yrs old… an apple fall to the ground made him wonder if gravity extended beyond Earth Developed a theory of universal gravitation Attractive force between two objects Watching Law of Universal Gravitation The force of attraction between any two masses is constant throughout the universe Sec. 8.2 Using the Law of Universal Gravitation m m F G A 2 B d problems involving orbital speed and period Relate weightlessness to objects in free fall Distinguish between inertia mass and gravitational mass Contrast Newton’s and Einstein’s views about gravitation 6.67 x 10-11 N·m2/kg2 If a projectile moves fast enough, it falls at the same rate that the Earth curves Objectives Solve G is a universal gravitational constant between two masses Satellite Motion The apple was also attracting the Earth Proposed Law of Universal Gravitation How fast are satellites moving? F = ma or F = mv2/r (ac = v2/r) F = G(mAmB/d2) Solve for velocity? Set them equal to each other G(mAmB/d2) = mv2/r which gives you…. 3 Period of a Satellite Circling Earth r3 T 2 GmE or if we know the velocity… T 2 r v Universal Law of Gravitation History Outline 1. Kepler used Brahe’ Brahe’s data to make Kepler’ Kepler’s Laws 2. Newton derived the universal law using Keplers Laws 3. Newton proved his law using the apple and the moon 4. Cavendish measures the universal constant Universal Law of Gravitation Newton derived the universal law of gravity He knew: a) T2/r3 = k b) v = 2 2r/T c) F = mv2/r Weightlessness What is gravity in outer space? Where space shuttle orbits…g = 8.7m/s2 How come astronauts are “floating” then? g = F/m Universal Law of Gravitation History Outline 1. Kepler used Brahe’ Brahe’s data to make Kepler’ Kepler’s Laws a) Measured the motion of the stars and planets b) Planets move from year to year but stars stay put c) Kepler developed laws to explain the motion of the planets Universal Law of Gravitation Newton derived the universal law of gravity He knew: a) T2/r3 = k b) v = 2r/T c) F = m( v )2 r 4 Universal Law of Gravitation Newton derived the universal law of gravity He knew: F = m42r a) T2/r3 = k T2 b) v = F = m42r c) F = m (2r/T )2 r3k r F= m42r2 F = 42 m T 2r k Universal Law of Gravitation Newton derived the universal law of gravity He knew: F = 42 m k But once he got to this part, he realized that every action had an equal and opposite reaction so, he had to add another “m” r2 F=G r2 m1 m2 r2 Universal Law of Gravitation F = Gm1m2 r2 F = the force of gravity between 2 objects Universal Law of Gravitation “G” was not known but the equation was still proven by the comparison of an apple and the Moon. m1 = mass of object #1 m2 = mass of object #2 Newton thought that maybe the Moon moved through the heavens for the same reason apples fell to the ground r = distance between their centers of mass G = Universal Gravitational Constant Universal Law of Gravitation If F = Gm1m2 r2 Then if F = ma then m1a = Gm1m2 r2 So, a = Gm2 r2 a = Gme r2 This equation works for any mass attracted to the Earth Universal Law of Gravitation 60re a = Gme r2 a= 9.81m/s2 re for an apple Since the Moon is 60x further away rmoon = 60re So, a = Gme (60re )2 amoon = Gme amoon = aapple 3600 3600re 2 5 Universal Law of Gravitation Universal Law of Gravitation amoon = .0027m/s2 according to the formula a = v2 The real acceleration of the Moon can be measured: a = v2 v = 2 2r/T T = 28.5days = 28.5x24x60x60 = 2462400s r = 60re = 60x 6.4x106m = 384000000m re = 6.4x106m r v = 2 2r/T T = 28.5days r = 60re Universal Law of Gravitation r Universal Law of Gravitation a = v2 a = v2 = (980m/s)2/384000000m = .0025m/s2 v = 2 2r/T v = 2 384000000m) / (2462400s) =980m/s 2384000000m) v = 2 2r/T v = 2 384000000m) / (2462400s) =980m/s 2384000000m) T = 28.5days = 28.5x24x60x60 = 2462400s r = 60re = 60x 6.4x106m = 384000000m re = 6.4x106m T = 28.5days = 28.5x24x60x60 = 2462400s r = 60re = 60x 6.4x106m = 384000000m re = 6.4x106m r Universal Law of Gravitation a = v2 = (980m/s)2/384000000m = .0025m/s2 r The real acceleration of the Moon = .0025m/s2 The theoretical acceleration = .0027m/s2 r Universal Law of Gravitation 1. Cavendish measures the universal constant A. “G” was still unknown for 100years B. Cavendish figured it out using a Torsion Balance This was so close that this became well accepted and Newton went down in history as the one who “discovered” discovered” gravity 6 Universal Law of Gravitation Torsion Balance 1. Imagine twisting the thread around 100 times Universal Law of Gravitation Torsion Balance 2. Then let go 2. Then let go 3. The system would spin in the opposite direction 3. The system would spin in the opposite direction Universal Law of Gravitation Torsion Balance 1. Now imagine the force that pulls the bar back Universal Law of Gravitation Torsion Balance 2. The force of a twisted wire is called torsion 3. Attach a spring scale to the bar and measure this torsion force F Universal Law of Gravitation Torsion Balance F 1. Imagine twisting the thread around 100 times 7. The force of a twist of 1/60th of a degree = .0144N / (100)(360)(60) = 0.00000000667N F 4. The force he measured was .0144N 5. Now he has to figure out how much torsion a tiny fraction of a twist would make. 6. If he twisted the string 1/60th of 1 degree, the force on the scale would be 100times 360 times 60 times smaller. Universal Law of Gravitation 8. He used a mirror attached to the string to reflect a beam of light onto a far away wall. 9. He used this to measure the angle the string had twisted Each marking measured 1/60th of a degree 7 Universal Law of Gravitation 10. 11. Universal Law of Gravitation He then placed a 1 kg ball at each end of the bar Next, he placed 1kg masses on the table near the masses on the bar 12. 13. 14. 15. Universal Law of Gravitation He let go. The force of gravity twisted the string 2/60th’s of a degree. The masses stopped moving. The force of gravity between the masses = the torsion in the string Universal Law of Gravitation Top View 12. 13. 14. 15. He let go. The force of gravity twisted the string 2/60th’s of a degree. The masses stopped moving. The force of gravity between the masses = the torsion in the string Torsion Force Fg 16. There were 2 forces from the two sets of balls 17. Total Torsion Force = 2Fg 18. The total angle it twisted was 2/60th’s of a degree Fg Torsion Force Universal Law of Gravitation Top View 16. Torsion Force = Universal Law of Gravitation Top View 22. m1 = = = 24. r 25. Fg = 0.00000000667N x 2 = 0.00000001334N 23. m2 Torsion Force 20. 2Fg = 0.00000001334N m1 = 1.0kg Fg 21. Fg = 0.00000000667N 7x1 r = .10m Fg = 6.6 Fg 0 -9 N Torsion Force m2 = 1.0kg 1.0kg 1.0kg .10m 0.00000000667N 26. Fg = Gm1m2 r2 27. G = Fg r2 m1m2 G = 6.67 x 10-11 Nm2 kg2 8 Universal Law of Gravitation F = Gm1m2 r2 F = the force of gravity between 2 objects m1 = mass of object #1 m2 = mass of object #2 Here’s a sketch of the Experiment from Cavendish’s time r = distance between their centers of mass G = Universal Gravitational Constant G = 6.67 x 10-11 Nm2 kg2 Sample Problems 1. No matter how much you say you don’ don’t find someone attractive, the fact is, that all people are at least gravitationally attractive. If you have a mass of 70kg and the other person has a mass of 80kg, what is the force of gravitational attraction between you both when you are sitting .50m apart? Sample Problem No matter how much you say you don’ don’t find someone attractive, the fact is, that all people are at least gravitationally attractive. If you have a mass mass of 70kg and the other person has a mass of 80kg, what is the force of gravitational gravitational attraction between you both when you are sitting .50m apart? Fg = ? m1 = 70kg m2 = 80kg Fg r = .50m G = 6.67 x 10-11 Nm2 kg2 9 Homework How the stars and planets move Pg 242 # 11-3 Pg 247 #3a,c, 5 North Pole Stars move in a circle as the Earth rotates on its axis every night. How the stars and planets move How the stars and planets move Also, the stars move from season to season through the year So to explain the motion of the stars, Moon and Sun you need to measure their positions at a certain time of day or night 10am 9am 11am 12am 1pm Night 2pm Day 8am 3pm 7am 4pm Summer night sky 6am Winter night sky 5pm How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky 10 How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky How the stars and planets move How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky How the stars and planets move Every year, at the same time, on the same day, the stars are in the same position they were last year. Night Summer night sky Winter night sky 12am 1/22/07 Day Winter night sky 11 How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky How the stars and planets move Summer night sky Winter night sky 12 How the stars and planets move How the stars and planets move Mars Summer night sky Winter night sky How the stars and planets move 12am 1/22/08 Winter night sky How the stars and planets move There are celestial objects that don’t stay in the same place – they wander. These are called “wanderers”, or in Greek, “Planets”. 12am 2/5/08 Winter night sky Winter night sky How the stars and planets move There are celestial objects that don’t stay in the same place – they wander. These are called “wanderers”, or in Greek, “Planets”. 12am 2/12/08 Winter night sky How the stars and planets move There are celestial objects that don’t stay in the same place – they wander. These are called “wanderers”, or in Greek, “Planets”. 12am 1/29/08 There are celestial objects that don’t stay in the same place – they wander. These are called “wanderers”, or in Greek, “Planets”. There are celestial objects that don’t stay in the same place – they wander. These are called “wanderers”, or in Greek, “Planets”. 12am 2/19/08 Winter night sky 13 How the stars and planets move How the stars and planets move There are celestial objects that don’t stay in the same place – they wander. These are called “wanderers”, or in Greek, “Planets”. Astronomers tried to explain the moving objects. At first, they thought these planets were like the Sun and the Moon. They thought the planets orbited around the Earth in a circle 12am 2/26/08 Winter night sky 12am 3/5/08 Winter night sky 14
