Expert: Philip Stahl - 1/22/2007

Question
Hi, I've really tried solving all of these question but I am really have problem solving them so can you please help me


33.   Measuring Masses. Mathematically. Newton’s version of Kepler’s third law reads:

P^2= 4pie^2/G(M1+M2)^a^3



where p is the orbital period in seconds, a is the average distance in meters, M1 and M2 are the masses of the two objects in kilograms, and the gravitational constant is

G=6.67x 10^-11 (m^3/kgxs^2)



Use this law to answer each of the following questions.

a.   The Moon orbits Earth in an average time of 27.3 days at an average distance of 384,000 kilometers. Use these facts to determine the mass of Earth. You may neglect the mass of the Moon and assume (    M earth  +  M moon  –  M earth       ).

b.   Jupiter’s moon Io orbits Jupiter every 42.5 hours at an average age distance of 422,000 kilometers from the center of Jupiter. Calculate the mass of Jupiter.
(     M Jupiter + M lo – M jupiter        )

a.   Use Earth’s orbital distance and orbital period to calculate the mass of the Sun.
(      M sun + M earth – M sun    )
 
d.   Pluto’s moon Charon orbits Pluto every 6.4 days with a semi major axis of 19,700 kilometer. Calculate the combined mass of Pluto and Charon. Compare this combined mass to the mass of Earth, which is about 6 X 10^24 kg.


Answer
Hello again,

This is a follow -up. It turns out that you can actually obtain the Earth's mass very nearly exactly in the first question - by simply using the units seconds (for P) and meters for a.

Thus, instead of using A.U. for a, simply convert the mean distance from 384,000 km to meters:

=  3.84 x 10^8  m

THIS then is the quantity that will be cubed

At the same time, convert the period of the Moon (27.3 days) to *seconds*, e.g.

27.3 days x (86400 secs/ day)

= 2358720  secs

THIS then is the quanitity to be squared (P^2)

Then simply substitute these values into your original eqn.

P^2= 4pi^2/G(M1+M2)^a^3

to solve for Earth's mass, viz.

->

M(E) =  4 (pi)^2/  G   x  [a^3/ P^2]


This same approach (using seconds for P, meters for a)  can also be used for the other problems in your set, and again - will give the more accurate answers than the original technique I suggested.  Just be sure you have a trusty calculator!