Force's and Newton's Laws of Motion

[ian_durrant]

Homework Statement

A spacecraft is on a journey to the moon. At what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the Earth balance that exerted by the moon? This point lies on a line between the centers of the Earth and the moon. The distance between the Earth and the moon is 3.85 × 108 m, and the mass of the Earth is 81.4 times as great as that of the moon.

Homework Equations

F= GM(e)m/r^2

The Attempt at a Solution

The only thing I've been able to come up with so far on this problem is to just set up a graph showing the two forces head to tail, i know that the moon's Force should be negative since it is attracting in the opposite direction of the earth? Any thoughts where to get started?

 

[Doc Al]

Imagine something at a distance X from the center of the earth. What force does the moon exert? The earth? Set up an equation to solve for X. (Draw a diagram for yourself.)

 

[Littlepig]

Don't use forces... My guess is using Potencial energy.
you know that in the point the potential energy=0

(well, you can use forces too, but this kind of problems can easily be solved by energy)

 

[Dick]

Don't use forces... My guess is using Potencial energy.
you know that in the point the potential energy=0

(well, you can use forces too, but this kind of problems can easily be solved by energy)

That's an amazing poor guess. Did you try working the problem out? Setting the potential energies (GM/r) equal gives you a different answer than setting the forces (GM/r^2) equal. Which do you believe?

 

[Littlepig]

ahah, my bad...Was confusing... thanks and sorry Dick.

 

[ian_durrant]

Ok I took the suggestion of solving for x and here's what I did:

x= distance from center of Earth where moon exerts more force

-F(1)= G81.4m/x (this is the force from the Earth, negative since its working in the opposite direction from the moon)

F(2)= Gm/(3.85*10^8-x) force that the moon is exerting

I then set the first equation equal to x and got x= (81.4mG)/-F(1) and plugged it into the other equation-

F(2)=Gm/[3.85*10^8-(81.4mG/-F(1))]

however the answer that i get does not make sense, and so i think I've messed up my inital equation somewhere- did I set up the x variable properly?

 

[Doc Al]

The general equation for the gravitational force between two objects is the one you quoted in your first post:

F = G m_1 m_2/r^2

Let m_1 be the mass of the moon (M) or Earth (81.4M) and let m_2 be the mass of the spacecraft (call it m since it doesn't matter).

r is the distance to the center of the moon or earth. Note that it is squared.

If we call the distance to the center of Earth X, what's the distance to the center of the moon? (What's the total distance between Earth and moon centers?)

Using that formula, find the point where the force from the Earth equals the force from the moon. Just set the magnitudes equal. (We know they point in opposite directions!)

Give it another shot.

 

[ian_durrant]

ahhh i see it now

ok i redefined variables:

F(m) = force from moon
F(e)= force from earth
G= gravitational constant
x= distance where force of moon is greater than earth
m(m)= mass of moon
m(2)= spaceship
Gives me...


F(m)= Gm(m)m(2)/x
F(e)= G81.4m(m)m(2)/ (3.85*10^8-x)^2

Now set those equal to each other...

Gm(m)m(2)/x= G81.4m(m)m(2)/ (3.85*10^8-x)^2

The G, m(m), and m(2) cancel each other out leaving me...

81.4x= (3.85 *10^8-x)^2

does that math look right?

 

[Doc Al]

ahhh i see it now

ok i redefined variables:

F(m) = force from moon
F(e)= force from earth
G= gravitational constant
x= distance where force of moon is greater than earth
m(m)= mass of moon
m(2)= spaceship
Gives me...

Good.

Note that the way you've used x below, it is the distance to the moon. You'll have to calculate the distance to the Earth once you solve for x. It's just 3.85*10^8-x.

F(m)= Gm(m)m(2)/x

Almost. This should be:
F(m)= Gm(m)m(2)/x^2
(Don't forget the square.)

F(e)= G81.4m(m)m(2)/ (3.85*10^8-x)^2

Good.

Now set those equal to each other...

Gm(m)m(2)/x= G81.4m(m)m(2)/ (3.85*10^8-x)^2

The G, m(m), and m(2) cancel each other out leaving me...

81.4x= (3.85 *10^8-x)^2

does that math look right?

Almost. You should end up with:
81.4x^2= (3.85 *10^8-x)^2

(Next you can take the square root of both sides.)