Lecture 9(+10) Physics 106 Spring 2006

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3/22/2006 Andrei Sirenko, NJIT 3 3/22/2006 Andrei Sirenko, NJIT 4 Lecture 9(+10) Physics 106 Spring 2006 Gravitation HW&R http://web.

main effect In the Galaxy (Milky Way): Attraction to the center of the Galaxy determines everything.

From that tim this problem became one of the frontiers of the modern Physics The Andromeda galaxy. Located 2.

1 3/22/2006 Andrei Sirenko, NJIT 3 3/22/2006 Andrei Sirenko, NJIT 4 Lecture 9(+10) Physics 106 Spring 2006 Gravitation HW&R Gravitation On Earth: the Earth gravitation dominates (F =mg) In the Solar System: attraction to the Sun is the main effect In the Galaxy (Milky Way): Attraction to the center of the Galaxy determines everything. At the edge of the Universe: the conceptual problems begin Accelerating expansion of the visible Universe is known since From that tim this problem became one of the frontiers of the modern Physics The Andromeda galaxy. Located light-years from us 3/22/2006 Andrei Sirenko, NJIT 1 3/22/2006 Andrei Sirenko, NJIT 2 Newton s Law of Gravitation (known since 1665) Measuring the Gravitational constant G using the Cavendish method

QZ9: Newton s Law of Gravitation Sample Problem from HW&R: m 1 = 6 kg, m 2 = m 3

. 2. What is the net gravitational force F 1 that acts on the particle 1 due to

M F shell = 0 M A uniform spherical shell of matter attracts a particles that is

Solid sphere is a combination of spherical shells: M 3/22/2006 Andrei Sirenko,

Gravitational acceleration: Gravitation inside the Earth Alice in Wonderland 1.

Mean Earth surface (0 km, 9.83 m/s 2 ), Mt. Everest (8.8 km, 9.

2 QZ9: Newton s Law of Gravitation Sample Problem from HW&R: m 1 = 6 kg, m 2 = m 3 = 4 kg, d 12 = a,, and d 13 = 2a. 2. What is the net gravitational force F 1 that acts on the particle 1 due to the other particles? Use a = 0.1 m. M F shell = 0 M A uniform spherical shell of matter attracts a particles that is outside the shell as if all the shell s s mass is concentrated at its center! Solid sphere is a combination of spherical shells: M 3/22/2006 Andrei Sirenko, NJIT 5 3/22/2006 Andrei Sirenko, NJIT 6 Gravitation Near Earth s Surface Gravitational acceleration: Gravitation inside the Earth Alice in Wonderland 1. Earth is not uniform. 2. Earth is not a sphere. 3. Earth is rotating. Mean Earth surface (0 km, 9.83 m/s 2 ), Mt. Everest (8.8 km, 9.80 m/s 2 ), highest manned balloon ( 36.6 km, 9.71 m/s 2 ), Space Shuttle orbit (400 km, 8.70 m/s 2 ), and communications satellite (35,700 km, m/s 2) 3/22/2006 Andrei Sirenko, NJIT 7 3/22/2006 Andrei Sirenko, NJIT 8

Gravitation inside the Earth Alice in Wonderland Outside the Earth (r > R) R = 6,400 km a g g = 9.

r = 5,400 km ma R = 6,400 km N ma g NYC (c) What is the speed of the train in the (b) What is the total force on the train in the middle

Answers: (a) v 6000 m/s (b) v 600 m/s (c) v 60 m/s (d) v 6 m/s (e) v 0 0 3/22/2006 Andrei Sirenko, NJIT 9 3/22/2006 Andrei Sirenko, NJIT

The Sun, of mass M,, is at one focus F of the ellipse. The other focus is F,, which is located in empty space.

3 Gravitation inside the Earth Alice in Wonderland Outside the Earth (r > R) R = 6,400 km a g g = 9.8 m/s 2 F K r ( G m 1 m 2 / R 3 ) r LA Gravitational Train Alice (a) Will this train move at all? r = 5,400 km ma R = 6,400 km N ma g NYC (c) What is the speed of the train in the (b) What is the total force on the train in the middle of the tunnel? (c) Where will the train stop? LA or in the middle of the tunnel middle of the tunnel? Answers: (a) v 6000 m/s (b) v 600 m/s (c) v 60 m/s (d) v 6 m/s (e) v 0 0 3/22/2006 Andrei Sirenko, NJIT 9 3/22/2006 Andrei Sirenko, NJIT 10 Planets and Satellites: Kepler's Laws A planet of mass m moving in an elliptical orbit around the Sun. The Sun, of mass M,, is at one focus F of the ellipse. The other focus is F,, which is located in empty space. Each focus is a distance ea from the ellipse's center, with e being the eccentricity of the ellipse. The semimajor axis a of the ellipse, the perihelion (nearest the Sun) distance R p, and the aphelion (farthest from the Sun) distance R a are also shown. 3/22/2006 Andrei Sirenko, NJIT 11 3/22/2006 Andrei Sirenko, NJIT 12

4 Planets and Satellites: Kepler's Laws Planets and Satellites: Kepler's Laws In time t,, the line r connecting the planet to the Sun (of mass M) ) sweeps through an angle θ,, sweeping out an area A (shaded). 3/22/2006 Andrei Sirenko, NJIT 13 3/22/2006 Andrei Sirenko, NJIT 14 Potential Energy : Potential Energy of a System : U U between r 1 and r 2 is the work done by the Gravitation Force during the move from r 1 to r 2 : r r 1 eference point r = 0 r r 3 r 2 U = 0; at infinity! (far away) 3/22/2006 Andrei Sirenko, NJIT 15 U = 0; at infinity! (far away 3/22/2006 Andrei Sirenko, NJIT 16

5 3/22/2006 Andrei Sirenko, NJIT 19 3/22/2006 Andrei Sirenko, NJIT 20 r Potential Energy : Is it U = mgh or, anyway a? It is the same thing, just different zero levels. h is more universal (always correct) U = mgh works for h << r, zero at the Earth surface always works, zero at U = GMm/r GMm/(r+h) = GMm(r+h-r)/(r (r+h)) = mh [GM/(r (r+h))] mgh 3/22/2006 Andrei Sirenko, NJIT 17 R Escape Speed: From energy conservation: E 1 = mv 2 /2 GmM/R E 2 = 0 (velocity is small) v 2 = 2GM/R= 2gR 3/22/2006 Andrei Sirenko, NJIT 18 v = (2GM/R) ½ 11,200 m/s First Satellite Speed: Newton s s cannon in 1687 in Principia Mathematica Potential Energy Satellites: Orbits and Energy Kinetic Energy for the orbital motion R v satellite (gr) ½ v satellite 8,000 m/s g 8.70 m/s 2 Total Energy: An object in orbit is weightless not because 'it is beyond the earth's e gravity' but because it is in 'free-fall' fall' - just like a skydiver.

6 Potential and Kinetic Energy Potential Energy Kinetic Energy for the orbital motion Total Energy 3/22/2006 Andrei Sirenko, NJIT 21 3/22/2006 Andrei Sirenko, NJIT 22

Lecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws

Lecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws Lecture 13 REVIEW Physics 106 Spring 2006 http://web.njit.edu/~sirenko/ What should we know? Vectors addition, subtraction, scalar and vector multiplication Trigonometric functions sinθ, cos θ, tan θ,