From Last Time Newton s laws. Question. Acceleration of the moon. Velocity of the moon. How has the velocity changed?

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1 Fro Last Tie Newton s laws Law of inertia F=a ( or a=f/ ) Action and reaction Forces are equal and opposite, but response to force (accel.) depends on ass (a=f/). e.g. Gravitational force on apple fro Earth is SAME AS gravitational force on Earth fro the apple. But acceleration of Earth due to this force is alost zero because of Earth s large ass. If an apple falls toward the Earth, why doesn t the oon fall toward the Earth? A. The oon is too big B. The oon is too far away C. The oon does fall toward the earth. Monday, Jan. 30 Phy107 Lecture 6 1 Monday, Jan. 30 Phy107 Lecture 6 Velocity of the oon What is the direction of the velocity of the oon? A C B Acceleration of the oon What is the direction of the acceleration of the oon? D. Zero accel C A B Monday, Jan. 30 Phy107 Lecture 6 3 Monday, Jan. 30 Phy107 Lecture 6 4 Acceleration = change in velocity change in tie How has the velocity changed? Velocity at tie t 1 Velocity at tie t 1 V(t ) V(t 1 ) Velocity at tie t Velocity at tie t Change in velocity Speed is sae, but direction has changed Velocity has changed Monday, Jan. 30 Phy107 Lecture 6 5 Centripetal acceleration = v /r, directed toward center of orbit. r = radius of orbit (In this equation, v is the speed of the object, which is the sae at all ties) Monday, Jan. 30 Phy107 Lecture 6 6 1

2 Earth s pull on the oon The oon continually accelerates toward the earth, But because of its orbital velocity, it continually isses the Earth. The orbital speed of the oon is constant, but the direction continually changes. Therefore the velocity changes with tie. True for any body in circular otion F= g F= g Experient Acceleration of ball 1 =F/ 1 = g/ 1 1 accelerates inward in response to force g Acceleration = v /r for circular otion Monday, Jan. 30 Phy107 Lecture 6 7 Monday, Jan. 30 Phy107 Lecture 6 8 A two-newton apple falls fro a tree. What is the net force on the apple? A. newtons downward B. 0 newtons downward C. 10 newtons downward I throw the N apple horizontally. After I throw it, what is the net force on the apple? A. 10 /s downward B. /s horizontally C. Need to know throwing force What is the acceleration? g~10 /s downward Monday, Jan. 30 Phy107 Lecture 6 9 This is because, after I throw the apple, gravity is the only force. The only acceleration is that due to gravity. Monday, Jan. 30 Phy107 Lecture 6 10, part I throw the N apple horizontally. After I throw it, what is the net force on the apple? N downward What is the acceleration of the apple? g~10 /s directly downward This is because, after I throw the apple, gravity is the only force. The only acceleration is due to gravity. Monday, Jan. 30 Phy107 Lecture 6 11 Shoot the onkey another exaple of superposition The dart gun is fired just as the onkey drops fro the tree. After the dart leaves the gun, the only force is fro gravity. The only deviation fro straight-line otion is an acceleration directly downward. The onkey has exactly the sae acceleration downward, so that the dart hits the onkey. Monday, Jan. 30 Phy107 Lecture 6 1

3 Newton s falling oon Throwing the ball fast enough results in orbital otion Fro Newton s Principia, 1615 Acceleration of oon The oon is accelerating at directly toward the earth This acceleration is due to the Earth s gravity. Is this acceleration different than g, the gravitational acceleration of an object at the Earth s surface? Can calculate the acceleration directly fro oon s orbital speed, and the Earth-oon distance. v r /s Monday, Jan. 30 Phy107 Lecture 6 13 Monday, Jan. 30 Phy107 Lecture 6 14 Distance and dia. of oon The radius of the earth The diaeter of the oon is the diaeter of its shadow during a solar eclipse. Fro the diaeter d and angular size d/r~5 deg, infer distance r~60*r(earth). Originally fro study of shadows at different latitudes by Eratosthenes R(earth)=6500 k Monday, Jan. 30 Phy107 Lecture 6 15 Monday, Jan. 30 Phy107 Lecture 6 16 Moon acceleration, cont Distance to oon = 60 earth radii ~ 3.84x10 8 Speed of oon? Circuference of circular orbit = "r orbital distance = "r Speed = =103 /s orbital tie = 7.3 days Centripetal acceleration = /s This is the acceleration of the oon due to the gravitational force of the Earth. Distance dependence of Gravity The gravitational force depends on distance. Moon acceleration is 9.81/s 0.007/s " 3600 ties saller than the acceleration of gravity on the Earth s surface. The oon is 60 ties farther away, and 3600=60 So then the gravitational force drops as the distance squared Newton: I thereby copared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth, and found the answer pretty nearly. Monday, Jan. 30 Phy107 Lecture 6 17 Monday, Jan. 30 Phy107 Lecture

4 Equation for force of gravity (Mass of object 1)#(Mass of object ) F gravity " square of distance between the F " 1 # For asses in kilogras, and distance in eters, Exaple Find the acceleration of an apple at the surface of the earth Force on apple = F apple = 6.7 "10-11 Earth " apple Acceleration of apple = F apple = 6.7 "10 #11 Earth apple This is also the force on the Earth by the apple d = distance between center of objects ~ radius of Earth F = 6.7 " " = 6.67 "10 #11 N # /kg " 5.98 "104 kg 6.37 "10 6 = 9.83 /s ( ) Monday, Jan. 30 Phy107 Lecture 6 19 Monday, Jan. 30 Phy107 Lecture 6 0 Gravitational force decreases with distance fro Earth Force on apple = F apple = 6.7 "10-11 Earth " apple So oving farther fro the Earth should reduce the force of gravity Typical airplane cruises at ~5 i = 8000 d increases fro 6,370,000 to 6,378,000 only about a 0.5% change International space station orbits at 350 k = 350,000 d = 6,370, ,000 = 6,70,000 Again d has changed only a little, so that g is decreased by only about 10%. Monday, Jan. 30 Phy107 Lecture 6 1 Monday, Jan. 30 Phy107 Lecture 6 So why is everyone floating around? The space station is falling siilar to Newton s apple Edward M. (Mike) Fincke, Expedition 9 NASA ISS science officer and flight engineer, is pictured near fresh fruit floating freely in the Zvezda Service Module of the International Space Station. (NASA) Jaes S. Voss, Expedition Two flight engineer, looks over an atlas in the Zvezda Service Module. (NASA) In its circular orbit, once around the Earth every 90 inutes, it is continuously accelerating toward the Earth at ~8.8 /s. Everything inside it is also falling (accelerating toward Earth at that sae rate). The astronauts are freely falling inside a freelyfalling elevator. They have the perception of weightlessness, since their environent is falling just as they are. Monday, Jan. 30 Phy107 Lecture 6 3 Monday, Jan. 30 Phy107 Lecture 6 4 4

5 A little longer ride Supree Screa feet of pure adrenaline rush A freefall ride d = 1 at t = d a Monday, Jan. 30 Phy107 Lecture 6 5 = " 300 ft 3 ft /s = 4.3 sec of freefall Parabolic path of freely falling object Monday, Jan. 30 Phy107 Lecture 6 6 Halfway to the oon, what is the acceleration of an apple due to the Earth s gravity? Moon is 60 Earth radii fro Earth) But gravitational force is only % weaker - Need to ove farther fro the Earth. A. g/ B. g/4 C. g/900 Moon is 60 Earth radii fro the Earth. Halfway is 30 Earth radii. So apple is 30 ties farther than when on surface. Gravitational force is (30) ties saller = g/900 Monday, Jan. 30 Phy107 Lecture 6 7 Monday, Jan. 30 Phy107 Lecture 6 8 Acceleration of gravity on oon On the oon, an apple feels gravitational force fro the oon. Earth is too far away. Force on apple on oon = F apple = 6.7 "10-11 oon " apple r oon Accel. of apple on oon = F apple = 6.7 "10 #11 oon apple r oon Copare to accel on Earth = 6.7 "10 #11 Earth r Earth accel. on oon accel. on Earth = oon / Earth ( r oon /r Earth ) Monday, Jan. 30 Phy107 Lecture 6 9 Accel. of gravity on oon accel. on oon accel. on Earth = oon / Earth ( r oon /r Earth ) = 7.4 "10 kg/6.0 "10 4 kg ( 1.7 "10 6 /6.4 "10 6 ) = (.65) = # 1 6 Monday, Jan. 30 Phy107 Lecture

6 Gravitational force at large distances: Stars orbiting our black hole At the center of our galaxy is a collection of stars found to be in otion about an invisible object. Orbits obey Newton s gravity, orbiting around soe central ass Scientists at the Max Planck Institute for Extraterrestrische Physik has used infrared iaging to study star otion in the central parsec of our galaxy. Movie at right suarizes 14 years of observations. Stars are in orbital otion about soe assive central object Monday, Jan. 30 Phy107 Lecture 6 31 Monday, Jan. 30 Phy107 Lecture 6 3 What is the central ass? One star swings by the hole at a iniu distance b of 17 light hours (10 A.U. or close to three ties the distance to Pluto) at speed v=5000 k/s, period 15 years. Fro the orbit we can derive the ass. The ass is.6 illion solar asses. It is ostly likely a black hole at the center of our Milky Way galaxy Monday, Jan. 30 Phy107 Lecture