How do the two different force formulae ($F = ma$ and $F = G\frac{m_1m_2}{r^2}$) relate to each other? $F = ma$ and $F = G\frac{m_1m_2}{r^2}$ are two formulae. Aren't they measuring the same thing? How do they relate to each other? 
 A: The first formula, $F=ma,$ describes the effect that forces have on objects, namely that a force causes an object to accelerate, with larger masses having slower acceleration.
The second formula, $F=\frac{Gm_1m_2}{r^2},$ describes the force generated by the gravitational attraction between two masses a certain distance apart.
In effect, the first formula tells you what forces do, and the second tells you how one kind of force is generated.
A: $\vec F = m \vec a$ is Newton's second law. It relates the net force acting on an object with the net acceleration produced by those forces. $\vec F = \frac{GMm}{r^2} \mathbf{\hat r}$ is Newton's law of universal gravitation. It tells us the attractive force by two masses towards each other. Since they were both developed by Isaac Newton, they are related, in a sense.
Consider two isolated masses, $m$ and $M$. The gravitational force are the only forces, so the net force also in the same direction. We can thus write $$ ma = -\frac{GMm}{r^2}$$ This is the force experienced by $m$ in a gravitational field of another mass $M$. Note that the negative sign is due to the attractive nature of the gravitational force. 
$a$ might thus be better represented as the gravitational field strength, $\vec g = -\frac{GM}{r^2} \mathbf{\hat r}$. This is the force per unit mass (in other words, the acceleration) experienced in a gravitational field - which makes sense, given that we canceled $m$ earlier.
A: When we use the same symbol for both forces, like here we use $F$ doesn't mean they represent the same thing. The $F$ in Newton's Second law or $F=ma$ represents the resultant force(vector sum of all forces which can be of any type like suppose gravitational or electromagnetic etc.) in any classical situation, for an object of mass $m$ to suppose calculate its acceleration $a$.  
The other formula $F=\frac{Gm_1m_2}{r^2}$ represents a special type of force which is gravitational force. It is the force of attraction between objects that have mass. Hence even is symbols are the same it is always important in physics to understand the context of the symbols. 
Note that these two formulas can become equal to each other if the only resultant force that is acting is just gravitational force. Then only you can write the two $F$s equal to each other. 
A: F=ma describes how a force will affect an object. For example, a force that yo exert to move a book.
The gravitational force formula gives you the force which occurs when two body with mass comes together. That is, it is only one kind of force. It tells you that any two bodies with mass will attract each other. 
In contrast, F=ma tells you that the more mass a thing has, the harder it is to accelerate it (or to make it stop moving).
They relate to each other simply by: mg=ma 
