Chapter 4 Gravitation

1. Chapter 4 Gravitation Physics Beyond 2000

2. Gravity • Newton • http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html • http://www.britannica.com/bcom/eb/article/9/0,5716,109169+2+106265,00.html • http://www.nelsonitp.com/physics/guide/pages/gravity/g1.html

3. Gravity • The moon is performing circular motion round the earth. • The centripetal force comes from the gravity. v Fc moon earth

4. Gravity • Newton found that the gravity on the moon is the same force making an apple fall. W Ground

5. F F m1 m2 r Newton’s Law of Gravitation • Objects attract each other with gravitational force. • In the diagram, m1 and m2 are the masses of the objects and r is the distance between them.

6. F F m1 m2 r Newton’s Law of Gravitation • Every particle of matter attracts every other particle with a force whose magnitude is G is a universal constant G = 6.67  10-11 m3kg-1s-2 Note that the law applies to particles only.

7. Example 1 • Find how small the gravitation is.

8. Shell Theorem • Extends the formula to spherical objects like a ball, the earth, the sun and all planets.

9. m2 m1 F F r Theorem 1a. Outside a uniform spherical shell. • The shell attracts the external particle as if all the shell’s mass were concentrated at its centre. O

10. Theorem 1b. Outside a uniform sphere. • The sphere attracts the external particle as if all the sphere’s mass were concentrated at its centre. m2 m1 F F O r

11. m1 F F O m2 r earth Example 2 Outside a uniform sphere. • The earth is almost a uniform sphere.

12. m1 m2 The two forces on m2 cancel. Theorem 2a. Inside a uniform spherical shell. • The net gravitational force is zero on an object inside a uniform shell.

13. m2 m1 F r Theorem 2b. Inside a uniform sphere. where m1 is the mass of the core with r the distance from the centre to the mass m2

14. m2 m1 F r Example 3 • Inside a uniform sphere.

15. Gravitational Field • A gravitational field is a region in which any mass will experience a gravitational force. • A uniform gravitational field is a field in which the gravitational force in independent of the position. • http://saturn.vcu.edu/~rgowdy/mod/g33/s.htm

16. F test mass m Field strength, g • The gravitational field strength, g, is the gravitational force per unit mass on a test mass. F is the gravitational force m is the mass of the test mass g is a vector, in the same direction of F. SI unit of g is Nkg-1.

17. F test mass m Field strength, g • The gravitational field strength, g, is the gravitational force per unit mass on a test mass. F is the gravitational force m is the mass of the test mass SI unit of g is Nkg-1.

18. field strength at X X M r Field strength, g, outside an isolated sphere of mass M • The gravitational field strength, g, outside an isolated sphere of mass M is O Prove it by placing a test mass m at a point X with distance r from the centre of the isolated sphere M.

19. Example 4 • The field strength of the earth at the position of the moon.

20. Field strength, g • Unit of g is Nkg-1. • g is also a measure of the acceleration of the test mass. • g is also the acceleration due to gravity, unit is ms-2.

21. Field strength, g. Unit Nkg-1. A measure of the strength of the gravitational field. Acceleration due to gravity, g. Unit ms-2. A description of the motion of a test mass in free fall. Field strength, g

22. Field lines • We can represent the field strength by drawing field lines. • The field lines for a planet are radially inward. planet Radial field

23. Field lines • We can represent the field strength by drawing field lines. • The field lines for a uniform field are parallel. Uniform field earth’s surface

24. Field lines • The density of the field lines indicates the relative field strength. g1= 10 Nkg-1 g2= 5 Nkg-1

25. direction of the force test mass Field lines • The arrow and the tangent to the field lines indicates the direction of the force acting on the test mass.

26. The earth’s gravitational field • Mass of the earth Me 5.98  1024 kg • Radius of the earth Re  6.37  106 m O Re

27. Gravity on the earth’s surface, go • The gravitational field go near the earth’s surface is uniform and The value of go  9.8 Nkg-1

28. Example 5 • The gravity on the earth’s surface, go.

29. Apparent Weight • Use a spring-balance to measure the weight of a body. • Depending on the case, the measured weight R (the apparent weight) is not equal to the gravitational force mgo. R mgo

30. Apparent Weight • The reading on the spring-balance is affected by the following factors: • The density of the earth crust is not uniform. • The earth is not a perfect sphere. • The earth is rotating.

31. Apparent Weight • The density of the earth crust is not uniform. • Places have different density underneath. Thus the gravitational force is not uniform.

32. Apparent Weight 2. The earth is not a perfect sphere. Points at the poles are closer to the centre than points on the equators. rpole < requator gpole > gequator N-pole Equator S-pole

33.  X Y Apparent Weight 3. The earth is rotating. Except at the pole, all points on earth are performing circular motion with the same angular velocity .However the radii of the circles may be different.

34.  Apparent Weight m 3. The earth is rotating. Consider a mass m is at point X with latitude . The radius of the circle is r = Re.cos  . X Y r Re  O

35.  Apparent Weight Fc m 3. The earth is rotating. The net force on the mass m must be equal to the centripetal force. X Y r Re  O Note that Fnet points to Y.

36.  Apparent Weight R R Fc 3. The earth is rotating. The net force on the mass m must be equal to the centripetal force. So the apparent weight (normal reaction) R does not cancel the gravitational force mgo. X Y r m mgo  O

37.  Apparent Weight R R Fc 3. The earth is rotating. The apparent weight R is not equal to the gravitational force mgo in magnitude. X Y r m mgo  O

38.  Apparent weight R on the equator mgo R The apparent field strength on the equator is

39.  Apparent weight R at the poles R mgo The apparent field strength at the poles is

40. Example 6 • Compare the apparent weights.

41.  Apparent weight at latitude  R Fc X Y r m mgo  O Note that the apparent weight R is not exactly along the line through the centre of the earth.

42. Me r g m O h Re Variation of g with height and depth • Outside the earth at height h. h = height of the mass m from the earth’s surface

43. Me m r g O h Re Variation of g with height and depth • Outside the earth at height h. where go is the field strength on the earth’s surface.

44. Me m r g O h Re Variation of g with height and depth • Outside the earth at height h. where go is the field strength on the earth’s surface.

45. Me m r g O h Re Variation of g with height and depth • Outside the earth at height h close to the earth’s surface. h<<Re.  where go is the field strength on the earth’s surface.

46. Me r O d Re Variation of g with height and depth • Below the earth’s surface. Only the core with colour gives the gravitational force. g r = Re-d

47. Me r O d Re Variation of g with height and depth • Below the earth’s surface. Find the mass Mr of g r = Re-d

48. Me r O d Re Variation of g with height and depth • Below the earth’s surface. g r = Re-d

49. Me r O d Re Variation of g with height and depth • Below the earth’s surface. g g  r r = Re-d

50. Variation of g with height and depth • r < Re , g  r. • r > Re , earth g go r distance from the centre of the earth 0 Re