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In Newton's second law F = ma, the mass is the object the force is being acted upon. But in a gravitational system, does that mean the mass of the object you are concerned with? For instance, if you are trying to figure out the acceleration of the Earth, do you only input the mass of the Earth into the equation?

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Yes, the force acting on an object at any instant is simply the mass of that particular object multiplied by the instantaneous acceleration that object is experiencing according to Newton’s 2nd law. The equations that describe the origin of a force, such as Newton’s law of gravity, will generally include all relevant objects that are interacting. In other words, Newton’s law of gravity will tell you the gravitational force acting between the two objects, which is equivalent to saying that it gives the force experienced by each object. As noted in a comment, it is also important that when doing such a calculation you know what reference frame that you are referring to when enumerating forces on a given object, as it can make the apparent force acting on an object seem different. You may be interested in the forces that each body is subject to in some reference frame (say, the rest frame of one of the bodies) in particular contexts. So if you look at the gravitational force in the frame of body 1, body 1 will be subject to no apparent force, while body 2 will accelerate towards it.

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You are correct. The equation $F=ma$ means that if you pick any object, and compute the forces acting upon that object, the mass of that object, and the acceleration that object is experiencing, then $F=ma$ will hold. All three simply need to refer to the same object to be consistent. If you are interested in the acceleration of the Earth, you need to use the mass of the Earth and the sum of the forces applied to the Earth.

Going one step beyond the question you asked, Newton's law of universal gravitation says that between any two objects, the magnitude of the gravitational force is $F=\frac{GMm}{r^2}$. If you make the assertion that the gravity of the sun is the only force acting on the Earth (ignoring solar pressures, gravitation attraction to Jupiter, etc.), then that $F$ is the sum of the forces acting on Earth. In that case, we get to equate the two equations, $\frac{GMm}{r^2}=ma$, so thus the acceleration of the Earth due to the sun will be $a=\frac{GM}{r^2}$ where $M$ is the mass of the sun, and $G$ is Newton's universal constant of gravity. If you consider any other forces, you can't equate the $F$ from $F=ma$ and the $F$ from $F=\frac{GMm}{r^2}$. One of them is the sum of all forces and the other is just one of them. It is unfortunate that both equations use the letter $F$ but with a different meaning.

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