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When a test mass enters the gravitational field of a larger mass, the test mass is attracted towards the larger mass as a gravitational force that is dependent on the strength of the gravitational field is exerted on it.

-> But how does the test mass exert an attractive force on the larger mass by Newton's third law?

Gravitational potential energy is the work done against gravity to bring a mass from infinity to some point, X. Really, it's gravity that's doing the work, which is why G.P.E becomes negative.

-> But what is the net gravitational potential energy at point X which is somewhere in between two masses?

-> I've seen the solution where the answer is to add the 2 values of gravitational potential energy as if the other mass weren't there. But why? What does net gravitational potential energy mean?

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Let's start with your second question. Looks like the root of misunderstanding is somewhere nearby.

But what is the net gravitational potential energy at point X which is somewhere in between two masses?

There is no such thing as "potential energy at point X". There is only a "gravitational potential". If you place a body of mass $m$ into a point with potential $V$, the potential energy of this body would be $m*V$.

There is a simple formula for gravitational potential at a given point. For each body you have to calculate (mass of the body)/(distance from the body to our point), sum up all the values and multiply the result by gravitational constant: $$V=-\sum{G*M_i/r_i}$$

Well, you have several bodies, you have calculated the potential at some point and you bring some new body of mass $m$ to that point. But wait, because of the new body the gravitational field and potential will change! The potential at our point will be different! Shouldn't we use the new/changed/correct value of potential at that point to calculate the potential energy of our body? No! To calculate the potential energy of some body we should calculate the potential (using the formula above) as if our body is not there.

Why is it so? We start with experimental fact that grav force between two bodies is $F=G*m*m_1/r^2$, and then mathematically prove that if $m_1$ is fixed than it's necessary to do some work to move the other body to infinity and the required work is $m*G*m_1/r$. If there are many bodies ($m$, $m_1$,...$m_k$), then the work required would be $m*(\sum{G*m_i/r_i})$. This is just a mathematical consequence of the fact that $F=G*m*m_1/r^2$. The portion of formula in parentheses we call potential (ehh, need to multiply by -1 first), it doesn't contain mass $m$ of the test body.

What does net gravitational potential energy mean?

"Net gravitational potential energy" is a property of a system of bodies. It is an amount of energy someone have to spend to move the bodies from far-far-away to a given position.

The formula is also quite simple. No energy is required to bring the first body from infinity. To bring the second body you have to spend the energy $-G*m_1*m_2/r_{12}$. To bring the third body: $-G*m_1*m_3/r_{13}-G*m_2*m_3/r_{23}$. And so on. Check that result doesn't depend on the order of the bodies.

But how does the test mass exert an attractive force on the larger mass by Newton's third law?

Because the smaller body also produces gravitational field. The grav force acting on the bigger body by gravitational field produced by smaller body is described by the same formula: $F=G*M*m/r^2$ It doesn't matter which mass in this formula is bigger or smaller.

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