1. 1 Examples and problems on gravitation

2. 2 The Earth and the Moon What is the magnitude of the gravitational force exerted by the Earth on the Moon? (m moon =7.36 x 10 22 kg, m Earth =5.98 x 10 24 kg, center-to-center distance from Earth to moon = 3.84 x 10 8 m)

3. 3 The Earth and the Moon LARGE!

4. 4 Weight on Earth Your weight is just the force exerted by the earth on you. What is the weight of an object with mass m on the surface of the Earth?

5. 5 Weight on Earth But we know that your weight is just equal to so

6. 6 Weight a distance d from the surface of the Earth What if mass m is a distance d from the surface of the Earth? (common scenario: airplanes, mountain climbing, etc.) Then the distance between the center of the Earth and the mass would change. Gravitational force on mass m, and hence, its weight will change!

7. 7 Weight a distance d from the surface of the Earth A mass m is a distance d away from the surface of the earth. So the weight of mass m (=gravitational force exerted by earth on mass m) is

8. 8 Weight a distance d from the surface of the Earth So the weight of mass m (=gravitational force exerted by earth on mass m) is SMALLER THAN THE WEIGHT ON THE SURFACE OF THE EARTH!

9. 9 Weight a distance d from the surface of the Earth m Weight varies inversely with the square of the distance from the earth’s center r= R E W(N) 700 500 300 0 200 100 400 600 m W= mg o r > R E r (x 10 6 m)

10. 10 Weight a distance d from the surface of the Earth Problem: At what distance above the surface of the earth is the acceleration due to gravity 0.980 m/s 2 if the acceleration due to gravity at the surface has a magnitude of 9.80 m/s 2 ?

11. 11 Weight a distance d from the surface of the Earth From Newton’s Law of Gravity: You can plug in the values of G, m Earth, R E, and solve for r’, but you also know that, at the surface of the Earth (1) (2)

12. 12 Weight a distance d from the surface of the Earth Divide (2) by (1): (3) (2)

13. 13 Weight a distance d from the surface of the Earth Solving for r’: But r’ is the distance from the CENTER (NOT the SURFACE) of the Earth to mass m: (3) (2)

14. 14 Weight Elsewhere The mass of Venus is 4.87 x 10ˆ24kg, and its radius is 6,051,000 m. (a) What is the acceleration due to gravity on the surface of Venus? (b) What is the weight of a 5.00 kg rock on the surface of Venus?

15. 15 Weight Elsewhere (a) What is the acceleration due to gravity on the surface of Venus? Plug in the data … and get

16. 16 Weight Elsewhere (b) What is the weight of a 5.00 kg rock on the surface of Venus?

17. 17 Multiple Objects Three particles are arranged as shown. What is the net gravitational force that acts on particle A due to the other particles? ParticleMass, kg mAmA 6.0 mBmB 4.0 mCmC mAmA mBmB mCmC 4.0cm 2.0cm FBAFBA FCAFCA

18. 18 Multiple Objects So what do we do with the forces???? We must add them as vectors!!! ParticleMass, kg mAmA 6.0 mBmB 4.0 mCmC mAmA mBmB mCmC 4.0cm 2.0cm FBAFBA FCAFCA

19. 19 Multiple Objects mAmA mBmB mCmC 4.0cm 2.0cm FBAFBA FCAFCA

20. 20 Multiple Objects Plugging in the numerical values, we get:

21. 21 Multiple Objects mAmA mBmB mCmC 4.0cm 2.0cm FBAFBA FCAFCA 76  104 

22. 22 Summary To apply Newton’s Law of Gravity: What are the masses involved? What is the distance between them? (remember, we need the CENTER-TO- CENTER distance) EVERYTHING’S HERE!

23. 23 Summary The acceleration due to gravity near the surface of the Earth

24. 24 How was the constant G measured? Click here to see how http://www.youtube.com/watch?v=4JG gYjJhGEE&feature=related

25. 25 Henry Cavendish was born on 10 October 1731 in Nice, France, where his family was living at the time. The Cavendish experiment, performed in 1797–98 was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant.[

26. 26 The apparatus constructed by Cavendish was a torsion balance made of a six-foot wooden rod suspended from a wire, with a 2-inch diameter 1.61-pound (0.73 kg) lead sphere attached to each end. Two 12-inch 348-pound lead balls were located near the smaller balls, about 9 inches away, and held in place with a separate suspension system. The experiment measured the faint gravitational attraction between the small balls and the larger ones.torsion balancelead

27. 27

28. 28 PlanetMass (kg)Radius (m) Mercury3.30 x 10ˆ232,440,000 Venus4.87 x 10ˆ246,051,000 Earth5.97 x 10ˆ246,378,000 Moon7.35 x 10ˆ221,738,000 Mars6.42 x 10ˆ233,397,000 Jupiter1.90 x 10ˆ2771,492,000 Saturn5.69 x 10ˆ2660,268,000 Uranus8.66 x 10ˆ2525,559,000 Neptune1.03 x 10ˆ2624,764,000 Pluto1.31 x 10ˆ221,160,000