Kepler's laws and weightlessness

Question

1. What will a spring scale read for the weight of a 55-kg woman in an elevator that moves

a) upward with constant speed of 6.0m/s,

b) downward with constant speed of 6.0m/s,

c) upward with acceleration of 0.33g,

d) downward with acceleration 0.33g and

e) in free fall?

2. The asteroid belt between Mars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroyed).

a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun? b) Can we use these data to deduce the mass of this planet?

Answer 1

I'm going to put weight in newtons by the way as this is the proper measure of it

Force = Mass x Acceleration

Reading = Mass x g +/- Mass x acceleration

+ for upwards, - for downwards

for a & b there is no acceleration so weight is just mass x gravity

a) and b)  539.55N

c) 717.6015N

d) 361.4985

e) 0

2)

a)

from keplers laws (see later for derivation) we can derive that the orbital period T and its distance R from the sun are proportional as below

T^2 proportional to R^3

so if R is 3 times bigger the T^2 = 27

hence T = square root of 27

this will give how many time longer it takes to orbit then the earth does

this works out at 1897.9 days

b) no we cant as the orbital period of a planet only depends on its distance from the sun and the mass of the sun as derived below

GMm/r^2   newtons law of gravity, M is mass of sun, m is mass of planet, r is distance from sun.

mv^2/r      centrifugal forve, v is planets velocity

equate these two

GMm/r^2 = mv^2/r    the m's cancel out as does one r, so we get

GM/r = v^2          converting from linear velocity to rotational velocity,

v^2 becomes 4 * pi^2 * r^2 / T^2  T is orbital period

equation becomes

GM/r = 4pi^2r^2/T^2

This can now be re arranged as

GM/r^3 = 4pi^2/T^2

Arrange again to get

(4pi^2/GM) * r^3 = T^2

as you can see only the mass of the sun is left