When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction to that of the first body.
Newton's third law is unintuitive to me. If the earth exerts a gravitational force on me, how do I exert a force on Earth?
Why is this true? It doesn't make any sense. How did Newton come to this conclusion?
Let's say you got fatter and fatter until you were the same mass as the earth, and, at the same time, the earth got smaller and smaller until it was only as massive as you were to start with. Now after this has happened, you must agree that you are now exerting a gravitational force on the earth (since you and the earth have now effectively switched roles, so since the earth was exerting a force on you before, you must now be exerting a force on the earth). But now at what point did you start exerting a force on the earth? How massive do you have to be before you can be considered to be exerting a force on the earth? The answer of course is that you were always exerting a force on the earth.
To see this mathematically we can look at the law for gravitation. This law tells us that if $A$ and $B$ are two objects with masses $m_A$ and $m_B$, and they are separated by a distance $d$, then the magnitude of the gravitational force of object $A$ on object $B$ is $F_{A \to B} = \frac{G m_A m_B}{d^2}$.
To calculate the force of the earth on you, consider the case where $A$ is the earth, which I will call $E$ and $B$ is you, which I will call $Y$. Then we compute the magnitude of the force of the earth on you to be $F_{E \to Y} = \frac{G m_E m_Y}{d^2}$.
To calculate the force of you on the earth, consider the case where $A$ is you, $B$ is the earth. Then we compute the magnitude of the force of you on the earth to be $F_{Y \to E} = \frac{G m_Y m_E}{d^2}$.
Now since $\frac{G m_E m_Y}{d^2} = \frac{G m_Y m_E}{d^2}$, we find $F_{E \to Y} = F_{Y \to E}$, that is, the magnitude of the force you exert on the earth is the same as the magnitude the force exerts on you. I assume you have no difficulty understanding that the directions will be opposite, since you are pulling each other towards each other.
We do exercise a force on Earth. Let's say that the force in question is F. When F acts on you it provides an acceleration of F/m (where m is your mass) and when the same force F acts on the earth it provides an acceleration of F/M (where M is the mass of the Earth). As M >>>>>> m, the acceleration provide does not create some perceptive change in the earth's moving (or anything else for that matter).
For an example, let's consider the following:
The total population of human being is, according to wolfram alpha, 7.13 billions of people. Let's assume an average weigh of 72 kg. This means a mass of 5.13 x 10 ^11 kg. The earth's mass is around 5.9 x 10^24 kg, still with wolfram alpha. The relationship m/M is on the order of 10^13 kg. That's how different the accelerations would be considering the entire population. Now imagine yourself. It's easy to see how unperceptive this is.
