What is the difference between F=G m1m2/r^ , F=Gm1m2*unit vector r/r^2, and F=-Gm1m2/r^2?

Question

I think some of these formulas are used for magnitudes of gravitational force and others just for gravitational force. I just wanted to be sure and need a clear explanation. Thank you in advance.

Answer 1

(i)

 

F = Gm₁m₂ / r²

 

This simply gives the magnitude of the gravitational force between m₁ and m₂.

 

(ii)

 

Let e_r = unit vector in the direction from m₁ to m₂.  The correct expression for the gravitational force on m₂ due to m₁ is

 

F₂ = -Gm₁m₂ e_r / r²

 

The negative sign simply means the gravitational force on m₂ due to m₁ is directed in the negative e_r direction, which is from m₂ to m₁.

 

(iii)  

 

F = Gm₁m₂ / r²

 

See explanation for (i).  It is the same.  Perhaps you meant

 

F₂ = -Gm₁m₂ r / r³

 

This is equivalent to (ii), since

 

e_r = r/r

 

and 

 

r / r³ = e_r / r²