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I'm wondering about some assumptions I have to make in deriving the gravitational potential energy. This arises from the following exercise: -enter image description here

Since the net force acting on the satellite is conservative, $$W=\oint_\mathcal{C}\vec F\cdot\mathrm{d}\vec r=0$$

Although I don't think that bit of information is necessary.

$$W = -GMm \ \int_{r_0}^{r_1} 1/r^2 dr$$

$$W = -GMm \left[-\frac{1}{r}\right]_{r_0}^{r_1}$$

Now, to arrive at the equation I'm looking to express, $r_0$ must equal $\infty$. However, I don't why this must be the case, to derive the potential energy. I know it's conventional to take the reference point of potential energy from an infinitely far away point, which is why $GPE$ is always negative, but it doesn't logically follow in the integral for me to do this, as it seems to imply the object was brought from $\infty$ to $r_1$ which doesn't make sense to me. So why is it that we set $r_0 = \infty$?

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Because the integral you write down is work, which is change in potential energy. If you want the potential at a point $r_1$, you need to calculate the work from the position $r_0$ where $V=0$. You can choose this arbitrarily by shifting the potential with a constant, but the problem specifies to take it at infinity.

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I think that your confusion stems from the fact that the problem that you were given asks you to calculate something that has no immediate practical value.

In a practical problem that, that is, a problem that asks you to predict the value of some quantity that you can measure, one always deals with differences or changes in the energy of a system. Knowing "the energy" of a particular system at a particular time is only useful if you know something about "the energy" at a later time. $\Delta E$. Or you may be given information about power or work and be asked to find $\Delta E$ (which could be used to answer questions about still-future events).

Constant additions always subtract themselves away. We may as well pick something convenient. "The potential energy" only has value because it can be compared to another instance of "the energy". By itself, it has no immediate value.

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