Universal Gravity?

Question

Find the point between Earth and the Sun at which an object can be placed so that the net gravitational force exerted by the Earth and the Sun on that object is zero.

Me=5.98x10^24
Ms=1.991x10^30
G=(6.673x10^-11)

F=(GM1M2)/r^2

Answer 1

Considering gravity only, you would just need to solve the equations that you have stated above.

f1 = G*Me*m/r1^2
f2 = G*Ms*m*r2^2

(subject to r1+r2 = distance from the earth to the sun )

f1 = f2

Me/r1^2 = Ms/r2^2

Defining R as the distance from the earth to the sun ( 1AU ) then the equation becomes:

Me/r1^2 = Ms/(R-r1)^2

solving for r1, I get r1=R/(1+sqrt(Ms/Me)) ~ R/578
which is about 259,000 km from the earth.

But if you include the fact that by orbiting the sun, one is in a rotational coordinate system, there are 5 points where one would have a net force of 0. These are called lagrange points. These points take into account the fact that along with the gravitational force, there is a centripetal "force" as well. So the equation becomes a little more complicated.

Answer 2

There are five Lagrange points where the Earth's and Sun's gravitational fields cancel out, but the one that's easiest to understand is the one in between the Earth and Sun. The mass of the Sun is 332776 times the mass of the Earth. The distance from the Earth to the Sun is 150 000 000 km., so this point is 1/332776 of this distance from the centre of the Earth. So it's 451 km. from the centre of the Earth in the direction of the Sun, which puts it deep underground.

Answer 3

Why exactly do you need help with this? You've written down the formula and the values of all the constants.... if you just set up an equation with Fsun on one side & Fearth on the other, the target mass cancels out, and you just have to figure out the necessary r's. To do it right, you also need to know the distance between the Earth & the Sun, and you really don't even need to know G. You get a system of equations:

Me * r(fromsun)^2 = Ms* r(fromEarth)^2

r(fromsun) + r(fromEarth) = Distance between sun & earth

Then just solve the system of equations.

Answer 4

Equate

Me/x^2 = Ms/(R-x)^2

where R is the distance between the center of the Earth and that of the Sun.

Answer 5

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