Physic 231 Lecture 21

Size: px

Start display at page:

Download "Physic 231 Lecture 21"

Rafe Whitehead
5 years ago
Views:

1 Physic 31 Lecture 1 Main points of today s lecture: Newton s law of universal gravitation: F GMm r PE GMm r Kepler s laws and the relation between the orbital period and orbital radius. T π GM 4 r 3

2 Key Concepts: Gravitation Newton s Law of Gravita1on Gravita1onal Accelera1on Planetary Mo1on Kepler s Laws Gravita1onal Poten1al Energy Conserva1on of Mech. Energy Ar1ficial Satellites Covers chapter 9 in Rex & Wolfson

3 Newton s law of universal gravitation All objects (even light photons) feel a gravitational force attracting them to other objects. This force is proportional to the two masses and inversely proportional to the square of the distance between them. F G 1 m m r 1 1 G x N m² kg²

A spaceship is on a journey to the moon. The masses of the earth and moon are, respectively, 5.98x10 4 kg and 7.36x10 kg. The distance between the centers of the earth and the moon is 3.85x10 6 m.

This point lies on a line between the centers of the earth and the moon. GMmoonmship GMearthmship Fmoon F earth Moon rmoon _ ship rearth _ ship GM m GM m multiply both sides by r earth_ship M moon!

4 A spaceship is on a journey to the moon. The masses of the earth and moon are, respectively, 5.98x10 4 kg and 7.36x10 kg. The distance between the centers of the earth and the moon is 3.85x10 6 m. At what point, as measured from the center of the earth, does the gravitational force exerted on the craft by the earth balance the gravitational force exerted by the moon? This point lies on a line between the centers of the earth and the moon. GMmoonmship GMearthmship Fmoon F earth Moon rmoon _ ship rearth _ ship GM m GM m multiply both sides by r earth_ship M moon! F ship _ Earth! rearth _ ship! rmoon _ ship! F ship _ moon Example Divide both sides by G Divide both sides moon ship earth ship rmoon _ ship rearth _ ship M moon M earth r earth _ ship r moon _ ship r earth _ ship r moon _ ship by m ship M moon M earth 4 rearth _ ship Mearth 5.98x10 kg moon _ ship moon 9.01 r M 7.36x10 kg r earth _ moon r earth _ ship + r moon _ ship 9.01r moon _ ship + r moon _ ship 10.01r moon _ ship 3.85x10 6 m Earth r 3.85x10 m r 3.47x10 m moon _ship 5 6 earth _ship

5 Example George weighs 100 N on the Earth. What would his weight be on the surface of another planet that has a planetary mass, which is 3 times the mass of Earth and mass and a planetary radius, which is times the radius of Earth? GMplanetmGeorge W GM m planet a) 75 N R WEarth planet R Earth b) 100 N c) 15 N GM planet m George 1 M p d) 150 N 3M E W planet W Earth Earth George R planet GM Earth m George R Earth Gm George M planet Gm George M Earth R planet 1 R Earth R Earth R Earth R planet R planet R p W E W p? R e 100 N W planet W Earth M planet M Earth R Earth R planet Hint: This is a ratio problem. W planet 3 4 W 3 100N 75N Earth 4

6 Conceptual question An astronaut floating weightlessly in orbit shakes a large iron anvil rapidly back and forth. She reports back to Earth that a) the shaking costs her no effort because the anvil has no inertial mass in space. b) the shaking costs her some effort but considerably less than on Earth. c) although weightless, the inertial mass of the anvil is the same as on Earth. 3

Kepler s laws Johannes Kepler proposed three laws of planetary motion: 1. All planets moved in elliptical orbits with the Sun at one of the focal points.

7 Kepler s laws Johannes Kepler proposed three laws of planetary motion: 1. All planets moved in elliptical orbits with the Sun at one of the focal points.. Planetary orbits sweep out equal areas in equal times. (This is a consequence of angular momentum conservation.) r Δθ area 1 r rδθ 1 r Δθ Δt Δt 1 r ωδt but r ω mr ω m Iω m L m const. 3. The square of the orbital period of any planet is proportional to the cube of the average distance from the planet to the Sun. For a circular orbit: GMstarmplanet mplanet v F grav m planetac r GM star v ω 3 r r r π ωt π ω GM T star π 3 4π r T T 4π T a3 For an elliptical orbit: GM star a GM star r 3 4

8 Example The Earth orbits the sun in an circular orbit of radius r E 1.5x10 11 m. What is the mass of the sun? 4π T r GM 3 sun 4π r M G T 3 sun ( ) ( 1.50x m) x 10 N m² kg² 365d 4h 3600s 1y y d d 30.0x10 kg 8

9 Satellite Orbits View from above North Pole, June Low orbit satellites: Hubble, space sta@on, Iridium system, spy sats GPS, research sats Geosta@onary communica@on sats

10 First cosmic speed First cosmic speed: speed of a satellite of mass m on a low-lying circular around a planet with orbit of M p and radius R p (g p GM p R p, r satellite R p ) GM m F ma m R E E v R I E GM R E E v I v I GM E R E GM E R E R E g R E For earth: g 9.81 ms R E 6.37x10 6 m v kms

Lecture 16. Gravitation

Lecture 16. Gravitation Lecture 16 Gravitation Today s Topics: The Gravitational Force Satellites in Circular Orbits Apparent Weightlessness lliptical Orbits and angular momentum Kepler s Laws of Orbital Motion Gravitational