Gravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition

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1 Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1

2 What you are about to learn: Newton's law of universal gravitation About motion in circular and other orbits How to calculate gravitational potential energy How to describe orbital types in terms of total mechanical energy The concept of escape speed The concept of gravitational field 2

Kepler s Observational Laws (1609 1619) 1.

A line segment joining a planet and the Sun sweeps out equal areas during equal

3 Kepler s Observational Laws ( ) 1. The orbit of a planet is an ellipse with the Sun at one of the two foci. 2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. 3

their separation. This law strictly holds only for point masses. However Newton showed that it applies to spherically symmetric masses.

4 Universal Law of Gravitation (1686) FF = FF 1 = FF 2 = GG mm 1 mm 2 rr 2 Newton's law of universal gravitation states that any two point particles attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of their separation. This law strictly holds only for point masses. However Newton showed that it applies to spherically symmetric masses. Also, it is a good approximation for any objects that are small compared with their separation. G is the constant of universal gravitation. 4

5 Cavendish s Experiment ( ) current value: G = (31) N m 2 /kg 2 5

6 Once G is known, then: FF = GG MM EEEEEEEEE mm 2 RR EEEEEEEEE = mm gg close to Earth s surface MM EEEEEEEEE = gg RR 2 EEEEEEEEE GG Earth s mass = ( ± ) kg RR EEEEEEEEE = m Earth s radius 3 gg ρρ EEEEEEEEE = 4 ππ RR EEEEEEEEE GG Earth s density = g/cm 3 6

7 Example 8.1 gg = GG MM rr 2 GG = N m 2 /kg 2 MM = MM EEEEEEEEE rr = RR EEEEEEEEE gg EE = GGMM EEEEEEEEE RR EEEEEEEEE 2 = 9.81 mm/ss 2 MM = MM EEEEEEEEE rr = RR EEEEEEEEE + h h = 380 kkkk gg h = GGMM EEEEEEEEE RR EEEEEEEEE + h 2 = 8.74 mm/ss 2 MM = MM MMMMMMMM rr = RR MMMMMMMM gg MM = GG MM MMMMMMMM RR MMMMMMMM 2 = 3.75 mm/ss 2 7

8 The Dependence on the Center-to-Center Distance of Spherical Objects 8

9 Closed Orbits are actually Ellipses 1 rr = bb + aa cccccc θθ 9

10 PhET 10

11 The Circular Orbit when the ellipse becomes a circle FF = GG MM mm rr 2 = mm vv2 rr vv = GG MM rr vv = 2 ππ rr TT TT 2 = 4 ππ2 rr 3 GG MM K3: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. 11

12 Example 8.2 h = 380 kkkk vv = GG MM rr = GG MM EEEEEEEEE RR EEEEEEEEE + h = 7.68 kkkk/ss TT = 4 ππ2 rr 3 GG MM = 4 ππ2 RR EEEEEEEEE + h 3 GG MM EEEEEEEEE = ss = 90 mmmmmm 12

13 PhET 13

14 Example: Period of the Moon rr = 384,402 km TT = 4 ππ2 rr 3 GG MM EEEEEEEEE = 27.3 days 14

15 PhET 15

16 Example 8.3 Geostationary Orbit: TT = 24 h Geostationary Operational Environmental Satellite TT 2 = 4 ππ2 rr 3 GG MM rr = GG MM EEEEEEEEE TT 2 4 ππ 2 1/3 = mm = kkkk h = rr RR EEEEEEEEE = kkkk RR EEEEEEEEE = km 16

17 Gravitational Potential Energy rr 2FFgggggggg WW gggggggg = rr. ddrr rr 1 = UU rr2 UU rr1 FF gggggggg rr. ddrr = FF gggggggg dddd FF gggggggg = GG MM mm rr 2 UU rr2 UU rr1 = GGGG mm 1 rr 2 1 rr 1 17

18 18

19 Example 8.4 mm = kkkk WW eeeeee = WW gggggggg = UU rr2 UU rr1 = GG MM EEEEEEEEE mm 1 1 rr 2 rr 1 = JJ rr 1 = RR EEEEEEEEE = km rr 2 = rr gggggggggggggg = kkkk WW eeeeee = WW gggggggg = UU rr3 UU rr2 = GG MM EEEEEEEEE mm 1 rr 3 1 rr 2 = JJ rr 2 = rr gggggggggggggg = kkkk rr 3 = rr mmmmmmmm = kkkk 19

20 A Simplifying and Meaningful Reference Point UU rr2 UU rr1 = GGGG mm 1 rr 2 1 rr 1 UU rr1 = GGGG mm rr 1 rr 1 = UU rr1 = 0 rr 2 rr UU rr = GGGG mm rr 20

21 Mechanical Energy (Kinetic + Potential Energies) EE = KK + UU = 1 2 mm vv2 GGGG mm rr 21

22 Example 8.5 vv oo = 3.1 kkkk/ss rr oo = RR EEEEEEEEE MM = MM EEEEEEEEE mm: unknown h =?? rr = h + RR EEEEEEEEE EE = KK + UU = 1 2 mm vv2 GGGG mm rr 1 2 mm vv oo 2 GGGG mm rr oo = 1 2 mm vv2 GGGG mm rr vv = 0 1 rr = 1 rr oo vv 2 oo 2 GG MM 22

23 Closed and Opened Orbits 23

24 Escape Velocity EE 0 EE = KK + UU = 1 2 mm vv2 GGGG mm rr 0 vv 2 GG MM rr 24

25 Gravitational Field close to the surface of a spherical object a generalized and more precise description 25

Work, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition

Work, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.