1. Thursday – review of Chapters 9-14 Today’s lecture –
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Second exam scheduled for Oct. 28th -- practice exams now available --
Thursday – review of Chapters 9-14
Today’s lecture –
Universal law of gravitation
Gravity near the planet’s surface
Gravitational potential energy
Planetary and satelite motion
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2. Newton’s law of gravitation: m2 attracts m1 according to:y m1
r2-r1
G=6.67 x N m2/kg2 m2 r1 r2 x
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3. Vector nature of Gravitational law:y m3 d x m1 m2 d
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4. Gravitational force of the EarthRE m
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5. Question:
Suppose you are flying in an airplane at an altitude of 35000ft~11km above the Earth’s surface. What is the acceleration due to Earth’s gravity?
a/g = 0.997
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6. Attraction of moon to the Earth:
Acceleration of moon toward the Earth:
F = MM a  a = 1.99x2020 N/7.36x1022kg = m/s2
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7. Stable circular orbit of two gravitationally attracted objects (such as the moon and the Earth)
REM F v a
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8. Peer instruction question
In the previous discussion, we saw how the moon orbits the Earth in a stable circular orbit because of the radial gravitational attraction of the moon and Newton’s second law: F=ma, where a is the centripetal acceleration of the moon in its circular orbit. Is this the same mechanism which stabilizes airplane travel? Assume that a typical cruising altitude of an airplane is 11 km above the Earth’s surface and that the Earth’s radius is 6370 km.
(a) Yes (b) No
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9. Stable (??) circular orbit of two gravitationally attracted objects (such as the airplane and the Earth) X
REa F v a
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10. Newton’s law of gravitation: Earth’s gravity:
Stable circular orbits of gravitational attracted objects: RE m
REM F a v MM
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11. More details
If we examine the circular orbit more carefully, we find that the correct analysis is that the stable circular orbit of two gravitationally attracted masses is about their center of mass. x
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12. Radial forces on m1: m2 R2 R1 m1 T2 ? Tangential forces ? 4/6/2019
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13. Peer instruction question
What is the relationship between the periods T1 and T2 of the two gravitationally attracted objects rotating about their center of mass? (Assume that m1 < m2.)
(A) T1=T2 (B) T1<T2 (C) T1>T2 m1 R2 R1 m2
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14. m1 R2 R1 m2
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15. What is the physical basis for stable circular orbits?
Newton’s second law?
Conservation of mechanical energy? E = K + U = (const)
Conservation of linear momentum? p = (const)
Torqued motion? t = I a ?
Conservation of angular momentum? L = (const)
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16. How are the magnitudes of L1 and L2 related?v2
L1=m1v1R1
L2=m2v2R2 v1
Question:
How are the magnitudes of L1 and L2 related?
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17. The potential energy associated with the gravitational force.
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18. Total mechanical energy for circular orbits: (assume M >> m) E2
U(J)
h/RE
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19. Peer instruction question
What is wrong with the previous analysis?
Nothing is wrong. (The description of circular motion due to gravitational attraction is complete.)
E depends on r and therefore must not be constant.
E can only be constant if r is constant, but it is not obvious why r is constant.
Conservation of angular moment will come to the rescue.
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20. Angular momentum: L = r x mv For circular orbit: L = RMEMMv
REM F a v MM
Angular momentum: L = r x mv
For circular orbit:
L = RMEMMv
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21. r E for circular orbit U(r) E for elliptical orbit 4/6/2019
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22. Circular orbit: y x Elliptical orbit: y x 4/6/2019
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23. Satellite h (km) T (hours) v (mi/h) Geosynchronous 35790 ~24 6900
Satellites orbiting earth (approximately circular orbits):
RE ~ 6370 km
Examples:
Satellite
h (km)
T (hours)
v (mi/h)
Geosynchronous
35790
~24
6900
NOAA polar orbitor
800
~1.7
16700
Hubble
600
~1.6
16900
Inter. space station*
390
~1.5
17200
*Link:
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24. Sample question:
Suppose that the space shuttle (m=105kg) was initially in the same orbit as the International space station (hi=390km) and the engines are fired to give it exactly the amount of energy DW to raise it to the same orbit as the Hubble space telescope (hf= 600km). What is the energy DW?
You can show that the energy of a satellite of mass m in a circular orbit of height h above the Earth’s surface is given by:
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