A newly discovered planet is 13.5 times more massive than the earth but half as dense. Approximately, how much longer would it take to orbit this new planet, and why is your answer an approximation?
Since density is mass/volume, the only way that the planet is 13.5 times more massive and yet half as dense is if it has 27 X the volume. If the planet is 27X more voluminous, and since it is probably spherical like the earth, it will have a radius that is three times(cube root of 27 = 3) larger.(the rest of terms cancel in the volume of sphere formula.) The circumference will also be three times larger, and at equal speeds, it will take approximately three times longer to orbit. The reason it is an approximation is that in reality you have to factor in the height above the surface. Let it equal to h, and let R = radius of earth. Cp= circumference of orbit around new planet;
Ce= circumference of orbit around Earth;
the ratio of the circumherences then becomes
Cp/Ce = 2p(3R +h)/[2p(R +h)]
If h is a lot smaller than R, then the answer will be approximately 3.
april 07 solution