OPENING QUESTIONS: We know that the potential energy of an object CLOSE to the Earth's surface is simply mgh.
However, we recently learned that the gravitational attraction by the Earth on an object decreases with the square of the distance from the Earth's center.
Reach back into the dim dark days of our learning about potential energy and recollect an equation that relates potential energy to a conservative force (like... um... I dunno, GRAVITY).
Now use that with what we recently learned about the universal gravitational force to derive an equation for gravitational potential energy that is DOES take into account the distance to the Earth's center.
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It's kind of surprising, the earth's orbit is very nearly circular with an eccentricity of only 0.0167 (according to Univ of Georgia Hyperphysics)
OBJECTIVE:
WORDS/FORMULAE FOR TODAY:
- "G" vs "g"
- Fg = Gm1m2/r2
Kepler's 1st: All planets move in ellipses with the sun at one focus.
Kepler's 2nd: Planet's orbits sweep out equal orbits in equal times:
- mathematically speaking that's dA/dt = constant
Kepler's 3rd: The (magnitude) of the square of the period of a planet's orbit is proportional to the (magnitude) of the cube of the (average) distance from the sun:
- Or in other words: T2α R3
- Or with the proportional constant: T2= KsR3
- Total Mech E of an orbiting body: E = -GMm/2r
WORK O' THE DAY:
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Let's get homework on the board please!
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Example 13.5 asks us to imagine that we are conducting studies of atmospheric phenomena in the upper atmostphere and we have to do those studies at a region above the Earth's atmosphere such that g is decreased by 1%.
Do that (without looking first if you please)
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Mr W discussion of geosynchronous orbits.... why do we care? Why are they useful?
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Take a look at section 13.6 and do example 13.7
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Mr W discussion of stellar evolution: white dwarfs, neutron stars & black holes (oh my!)
HOMEWORK:
Chapter 13 problems begriming on page 412:
Problems 31 & 33 (see additional questions for 33 below)
- (please make the following ADDITIONS to 33 part b:
- determine the surface acceleration if the sun were to become a neutron star (radius = 10km) as well (it's not massive enough to do that but it is an interesting "what if"
- similarly the sun does not have enough mass to become a black hole. But we can do the math and determine that if the sun somehow WERE to become a black hole it would have a radius of 3 km. What does THAT tell us about the gravitational acceleration a the Event Horizon of a black hole?
- Does your answer make sense? Let's be sure and discuss
- Part c: Why CAN'T we use mgh?