Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1
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. 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
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
Cavendish s Experiment (1797 1798) current value: G = 6.67408(31) 10 11 N m 2 /kg 2 5
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 = (5.9722 ± 0.0006) 10 24 kg RR EEEEEEEEE = 6.3781 10 6 m Earth s radius 3 gg ρρ EEEEEEEEE = 4 ππ RR EEEEEEEEE GG Earth s density = 5.5153 g/cm 3 6
Example 8.1 gg = GG MM rr 2 GG = 6.67 10 11 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
The Dependence on the Center-to-Center Distance of Spherical Objects 8
Closed Orbits are actually Ellipses 1 rr = bb + aa cccccc θθ 9
PhET 10
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
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 = 5.52 10 3 ss = 90 mmmmmm 12
PhET 13
Example: Period of the Moon rr = 384,402 km TT = 4 ππ2 rr 3 GG MM EEEEEEEEE = 27.3 days 14
PhET 15
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 = 4.22 10 7 mm = 42.2 10 3 kkkk h = rr RR EEEEEEEEE = 36.0 10 3 kkkk RR EEEEEEEEE = 6.4 10 3 km 16
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
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Example 8.4 mm = 11 10 3 kkkk WW eeeeee = WW gggggggg = UU rr2 UU rr1 = GG MM EEEEEEEEE mm 1 1 rr 2 rr 1 = 5.842 10 11 JJ rr 1 = RR EEEEEEEEE = 6.4 10 3 km rr 2 = rr gggggggggggggg = 42.2 10 3 kkkk WW eeeeee = WW gggggggg = UU rr3 UU rr2 = GG MM EEEEEEEEE mm 1 rr 3 1 rr 2 = 0.925 10 11 JJ rr 2 = rr gggggggggggggg = 42.2 10 3 kkkk rr 3 = rr mmmmmmmm = 384.4 10 3 kkkk 19
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
Mechanical Energy (Kinetic + Potential Energies) EE = KK + UU = 1 2 mm vv2 GGGG mm rr 21
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
Closed and Opened Orbits 23
Escape Velocity EE 0 EE = KK + UU = 1 2 mm vv2 GGGG mm rr 0 vv 2 GG MM rr 24
Gravitational Field close to the surface of a spherical object a generalized and more precise description 25