sample determine the net gravitational force acting on the earth during a total lunar eclipse. m sun...
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SampleSampleDetermine the net gravitational force acting on the Earth during a total lunar eclipse.
msun = 1.99 x1030 kg
mmoon = 7.36 x 1022 kg
rsun = 1.5 x 108 km
rmoon = 384 400 km
Kepler (1571-1630)
Used Tycho Brahe's precise data on apparent planet motions and relative distances.
Deduced three laws of planetary motion.
Took him the last 30 years of his life.
Kepler’s First Law• The orbits of the
planets are elliptical (not circular) with the Sun at one focus of the ellipse.
• 'a' = semi-major axis: Avg. distance between sun and planet
Kepler’s First Law• Perihelion – close to
sun (perigee)
• Aphelion – furthest from sun (apogee)
• Eccentricity – how not a circle are you?
• circle e = 0
• parabola e = 1
Kepler’s First LawExamples:
Earth: e = 0.0167Mercury: e =
0.2056Venus: e =
0.00657
Kepler’s First Law
a = semi-major axis
Kepler's Second Law
A line connecting the Sun and a planet sweeps out equal areas in equal times.
Translation: planets move faster when closer to the Sun.
slower faster
Kepler's Second Law
A line connecting the Sun and a planet sweeps out equal areas in equal times.
slower faster
The speed of the planet in orbit is dependent on its distance from the sun
voro = vf rf
SampleSampleIf the Earth has an orbital speed of 29.5 km/sec at apogee, determine the orbital speed at apogee.
ra = 1.52 x 108 km
rp = 1.47 x 108 km
Kepler’s Second LawKepler’s Second Law
voro = vf rf
Note where the highest speeds of tornados
and hurricanes are.
Kepler’s Third LawKepler’s Third LawThe square of a planet’s orbital period is proportional to the cube of its semi-major axis
.
Translation: the further the planet is from the sun, the longer it will take to go around
Why Does it Work?Why Does it Work?
Newton discovers that ellipses are pretty close to circular, just with the sun offset
Fc = Fg
SampleSampleHow fast must a satellite move to maintain an orbit of 500 km over the Earth’s surface?
What is the period of rotation?
Kepler’s Third LawKepler’s Third Law
Newton took the idea of centripetal force and applied it to the Kepler problem
Fc = Fg
SolutionSolution
.
Where M is the mass being orbited (as opposed to orbiting)
T is the period of the orbit
r is the radius of the orbit