6. Summarize Newton s Law of gravity and the inverse square concept. Write out the equation

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1 HW due Today. 1. Read p Summarize the historical account of Brahe and Kepler 3. Write out Kepler s 3 laws. 1) Planets in orbit follow an elliptical path, the Sun is located at a focus of the elliptical path. 2) A line connecting the planet to the Sun will "sweep out" = areas in =times 3) T 2 a r 3 a = T 2 b r 3 b 4. Do reviewing concepts 1 3 and appl. Conc 16 on p RC1: The path of Io is elliptical; Jupiter is located at one of the foci of that elliptical path. RC2: The earth is closer to the sun in the N hemisphere's winter RC3: No, the 2nd law applies to each planet individually AC16: 5. Read p Summarize Newton s Law of gravity and the inverse square concept. Write out the equation Newton found that the force of gravity decreases both inversely quadratically as objects get farther apart. F g = G m 1m 2 d 2 Period 4 Egg launches: prototype This Thursday Dec. 13 Final: Wednesday Dec.19 Period 8 Egg launches: prototype This Friday Dec.14 Final: Thursday Dec.20 Chapter 8 Kepler's Laws and Universal Gravitation Isaac Newton

2 1500+ years of not caring re sun centered solar system about analytical science Copernicus 1500 late 1500's Aristarchus ~230 BCE Kepler Brahe 1500's Copernicus proposes a new heliocentric model Copernicus' model has issues culturally and mechanically Tycho Brahe late 1500's Most advanced and precise astronomical equipment & data Does not support Sun at center of solar system after his dueling mishap

3 Johannes Kepler early 1600's uses the decades of data accumulated by Brahe to develop clear mathematical laws of planetary motion. Kepler does this without any notion of gravity forces.

4 Kepler's First Law of Planetary Motion 1. The paths of the planets around the Sun are elliptical in shape: the Sun is located at one of the foci of that ellipse. Sadly, the other focus is... empty 30 days 30 days 30 days Kepler's 2nd Law of planetary motion An imaginary line connecting the planet to the sun will sweep out equal areas in equal times. This tells us that the satellites(planets) move faster when closer

5 Kepler's 3rd Law: Summary of 3rd law: All the planets orbiting the same object: The ratio of their periods squared to their orbital radius cubed is a constant T 2 a r 3 a = T 2 b r 3 b or T a 2 r a 3 = T b 2 r 3 b

6 Planet Orbital Period, T T 2 Average Orbital Radius, r r 3 T 2 /r 3 units > planet Earth years Mercury Years squared meters meters^3 Yrs^2 m^ E E E 34 Venus E E E 34 Earth E E E 34 Mars E E E 34 Jupiter E E E 34 Isaac Newton!

7 Newton proposes gravity is a... UNIVERSAL Force! F g = G m 1m 2 d 2

8 Isaac Newton!

9 Newton proposes gravity is a... UNIVERSAL Force! " 'tis the force of gravity that doth provides the circular impetus" Newton came up with the equation, but not the size of G

10 " Alas, I am smart" Newton came up with the equation, but not the size of G Henry Cavendish did a painstaking experiment to determine G (the universal gravitational constant)

11 CAVENDISH experiment!

12 What is the force of attraction between Justin Bieber and One Direction????? distance between centers one direction mass = 240 kg 6.0 meters Bieber mass = 40 kg

13 New HW : handout Egg is due SOON!!! period 5 tomorrow Thursday period 3 Friday Two masses are 3 meters apart and their gravitational attraction is 12 Newtons 3 meters What is the new gravitational force if the distance is doubled to 6 meters? new r 6 meters

14 1.5 m 4.5 m F g New F g? 1.5 m.75 m F g 1) so, first we see what changed: new F g? the distance (the r) is reduced by 1/2 2) then, we put the 1/2 in front of the r and see what happens... 3) go through the math to see tha F g by increasing by a factor of 4

15 if... What happens to F g???? both masses are doubled and the r between them is doubled... The effects cancel There's NO change one mass is tripled and the distance (r) between them is halved... This increases the F g by a factor of 12 both masses are halved and the distance (r) between them is tripled... This reduces the F g by a factor of 1 36 Big G versus little g

16 What is it? g Acceleration due to gravity near the Earth's surface how many m/s the velocity of a free falling object changes by each second vs. What is it? G The Universal Gravitational Constant It is an relative indication of how strong(weak) gravity in our universe does it vary? Yes! It's a "local" value different on Moon, Jupiter etc. does it vary? No! It is universal, the same everywhere! How can you find weight with it? F g = mg What is its value? 9.81 m/s 2 ( just near the Earth's surface) How can you find weight with it? What's its value? Fg = G m 1m 2 G = 6.67 x 10 r 2 11 Nm 2 kg 2 everywhere! So now we know G, so what??? how to figure out the mass of your planet???? Up until Cavendish determined the value of Big G, we were not sure about what the mass of the Earth was ( we weren't sure of its density) So know we could take advantage to the two different ways to find weight (F g ) and set them equal: 5.9 x kg

17 To do these types of problems: Say #3 1) write out the F g equation : Fg = G m 1m 2 r 2 2) re write it with space for changes Fg = G m 1 m 2 new ( r) 2 3) Commute out the changes and see what occurs to the original equation Fg = G m 1m 2 new r 2 4) so, the new force is 4x the original force... 4 x 60 N = 240 N To do these types of problems: Let's look at #5 1) write out the F g equation : Fg = G m 1m 2 r 2 2) re write it with space for changes Fg = G m 1 m 2 new ( r) 2 3) Commute out the changes and see what occurs to the original equation Fg = G m 1m 2 new r 2 = G m 1m 2 = G m 1m 2 r 2 r 2 4) so again, the new force is 4x the original force... 4 x 60 N = 240 N

18 Satellites Satellites can be thought of as in a "special kind of freefall". They go fast enough so that they fall away from the straight path at the same pace the earth's surface curves away from "flat". Weight force is acting as Centripetal Force flat F g tangential velocity

19 Relationship between orbital radius and acceleration due to gravity (g): Set F g = mg equal to new F g at an altitude of 1000 km? g = 9.81 m/s 2 near surface So, the acceleration of gravity varies just like weight does... If you double your distance from a planet, the accel. of gravity is quartered

20 Relationship between orbital radius and satellite velocity: Set F c equal to new F g We can see that as a satellite's orbital radius is increased, the speed needed to maintain orbit decreases This also means that the orbital period (time for one "lap"), increases International Space Station speed Speed of a communication satellite Speed of the moon 7800 m/s 3100 m/s 1100 m/s altitude above surface 410 km 3500 km km (avg)

21 We use that equation to find the mass of other things in space... As long as something orbits it and we measure the satellite's speed, we can find the mass of it altitude above surface (avg)

22 Chapter 8 day notebook December 15, 2015 Geosynchronous orbit A satellite in geosynchronous orbit has 2 requirements: 1) the orbital radius is the correct size so that one complete orbit takes 24 hours (one earth rotation) 2) the orbit path of the satellite lies in the plane of the Earth's equator

23 A geosynchronous satellite remains above the same spot of the Earth all the time. It must orbit at a unique distance in the same direction as the Earth's rotation, in the plane of the earth's equator if time convert to T =...

24 Satellite locations Lots near the Earth Geosynchronous radius

25 Chapter 8 end tides and more.notebook December 16, 2015 g gravitational field strength is an indication of how strong gravity is at a particular location g is measured in units of Newtons kg = N kg Here, at the surface of the Earth, g is 9.81 As distance from the Earth's surface increases, g decreases gravitational field strength g distance from planet

26 Chapter 8 end tides and more.notebook December 16, 2015 determine the gravitational field strength F g = N F g 8.00 kg rock

27 I AM A GRAVITON! I AM GRAVITON! I AM A GRAVITON! I AM GRAVITON! I AM A GRAVITON! I AM GRAVITON! I AM A GRAVITON! I AM A GRAVITON! I AM GRAVITON! I AM A GRAVITON! I AM A GRAVITON! I AM A GRAVITON! I AM A GRAVITON! I AM A GRAVITON! Chapter 8 end tides and more.notebook December 16, 2015 An astronaut whose weight on the Earth's surface is N travels to the surface of a different planet where her weight force is N. What is the gravitational field strength on the surface of this planet? Gravitational Models and Theories 1) Gravitational Field and Newton's Gm 1 m 2 r 2 g = F g m 2) Graviton particle exchange model MASS 1 MASS 2 GRAVITONS 3) Einstein's General Theory mass "warps" space

28 Chapter 8 end tides and more.notebook December 16, 2015

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