Gravity/Planetary Forces Question - High School physics question

[closertolost]

Homework Statement

Find the distance an object needs to be in between the sun and the Earth for it to be perfectly balanced (not moving)

Homework Equations

mass of Earth = 5.98e24 kg
mass of sun=1.991e30 kg
distance between sun and Earth = 1.479e11 m
Fg=Gm₁m₂/r²

The Attempt at a Solution

well, i tried making the Fg zero for two equations - one with Fg of the object to the sun, and one with Fg of the object to the earth..but it got messed up..can someone help me?

can i sub in the mass of the object as 1kg?

 

[cepheid]

Hint: The point at which the object won't accelerate is the point at which the gravitational forces on the object are *equal* in magnitude (and opposite in direction).

No, you don't need to substitute in a numerical value for the mass of the object, because it should cancel from both sides of the equation (meaning that the point at which the forces balance doesn't depend upon how heavy the object is).

 

[closertolost]

so, do i put
Fg=(6.67e-11 x 5.97e24)/d²
Fg=(6.67e-11kg x 1.991e30kg)/1.479e11m-d² <---because d is the distance to the object, and 1.479e11 is the distance from sun to Earth (?)

 

[closertolost]

oh, and then cram the equations together :P

 

[cepheid]

closertolost

When I hinted that the gravitational forces on the object due to Earth and the sun were equal, I meant for you to actually *equate* them. So, using the subscripts E and S for Earth and sun respectively, we have:

F_E = F_S

GmM_E/r² = GmM_S/(R-r)²​

You'll notice I've done something interesting here with the distances. I've decided to call the distance between the Earth and the sun "R." So, if the object is a distance r from Earth, then its distance from the sun must necessarily be R-r (since the problem states that the object lies along a straight line connecting the two celestial bodies). If you're still not sure, draw a diagram. EDIT: I see that you already figured this out, nice work.

You'll also notice that m, the mass of the "test object" cancels from both sides of the equation, and so does the gravitational constant, G. This leaves you with an equation for r (the thing you are trying to calculate) in terms of three known quantities. The three known quantities are the masses of the Earth and sun, and the Earth-sun distance, R.

 

[closertolost]

ahah thanks very much! i have it now :)