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haha
upon careful observation of planetary motion
scientist realize that the planets have common regularities
center has kaplan an astronomer and mathematician
stated these regular tease in his laws of planetary mission
cap list first law is that each plant has analytical orbit
with the sun at one focus
an ellipses a plane figures such that the some of the distances from any point
on the figure to to fixed points in the plane is constant
each fixed point is called a focus
the longer lines segmented the foci is the major access
half of that is called the semi major axis and perpendicular to the major axis
going to be ellipses midpoint is the minor axis remember this
well couplers second law states an imaginary line that connects the sun to
any planets sweeps out equal areas in equal times
cappers third law and will probably use this a lot sees that the square the
planet's orbital period is proportional to the cuban the semi major axis
remember we learned about it objects period in the crab impact will that
there's ex parte one
that period capital t_ is the time it takes for object to complete one orbit
that period measured in seconds is how long it takes for planet to complete one
orbit around the sun
or we can use it for any object that orbits around another like two
satellites
so popular spirit law states that the square than objects orbit around another
object
it was a constant times that orbiting objects any major access killed
looking like this as an equation
where the concert has been found to be four pi squared
over the gravitational constant times the mass of the body around which the
object organs
for us ten
t squared equals four pi squared over g times and
times eight cubed where t is the period of the planet g_e_ is the gravitational
constant
and is the mass about which the object orbits and eight is the orbiting objects
semi major axis or it's orbital radius
got that
t squared equals four price squared over g_m_ tie-ins eight q
let's try this thing in a problem
tightness
a mood of saturn has an orbital radius around saturn of one point two two times
ten to the ninth meters
and saturn has a massive five point six times ten to the twenty six kilograms
so how many days does it take for tightened to complete one orbit around
saturn
first let's figure out what the question is asking us to find
it's asking us how many days it takes a moment of saturn
to complete one orbit around saturn well that's the period t let's look at our
equation we've been given the mass of saturn amma and we know the
gravitational constant g right and the problem gives us pay because that's the
orbital radius of tightening which is also the length of it semi major axis so
now we can crank out an answer
ke squared equals four pi squared over gene and times little a cubed
filling in our values we get t squared equals four pi which will approximate as
three point one for squared over the gravitational constant six point six
seven times ten to the negative eleventh newton's times meter squared over
kilogram squared times the mass of saturn five point six times tend to the
twenty six kilograms
all times the semi major axis one point to to times ten to the ninth meters
cubed
let's do the numbers then figure out the units
starting over here and cuba in one point two two times ten to the ninth we get
one point eight two times ten to the twenty seventh
squaring three point one four in the numerator and multiplying by four we get
thirty nine point four
multiplying this by the number which has gone on the outside we'd get seventy one
point seven one times tend to the twenty seventh
in the denominator six point six seven times ten to the negative eleven times
five point six times ten to the twenty sixth is thirty seven point three five
times tend to the fifteenth
dividing these two answers we get the qty squared equals one point nine two
times ten to the twelfth
how about the units
well we have meters cubed over nunes times meter squared over kilogram
squared times kilograms
converting news to kilogram meters over second squared we can start canceling
things out
the leaders all cancel out as well as the kilograms so we just have one over
second squared in the denominator which flips over to leave us with second
squared as the unit for our answer
now we're not quite done because this is t squared and we're looking for tea
so taking the square root we get one point three nine times ten to the sixth
seconds
so titans period is one point three nine times tend to the sixth seconds now i
cannot checklists happy question properly
one little tactics if you look at the index at the units cancel out end up
with just seconds you get all right
but wait we're not done yet
we just found out titans period one point three nine times canada six
seconds but the problem asks for us define titans period in case
we don't titans period
is one point three nine pans tend to the six seconds
there are sixty seconds in one minute
sixty minutes in one hour and twenty four hours in one day
in order to convert one point three nine times came to the six seconds in two
days
multiply by all the ratios
and we get sixteen point one days
so it takes pride in sixteen point one days to make one orbit around sat
and to watch more premeditation abuse and be there website
dr