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zenterix
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- Homework Statement
- While re-reading a chapter on potential energy and conservation of energy, I noticed that there are some conceptual questions I still have about potential energy.
- Relevant Equations
- The chapter I am reading has the following quote
When only internal conservative forces act in a closed system the sum of the changes of the kinetic and potential energies of the system is zero.
Is gravitational potential energy defined only for internal conservative forces?
For reference, this is the chapter.
Suppose we have two objects: object 1 and also object 2.
If we consider the system to be both objects, then we can compute the work done by the pair of internal conservative forces
$$\int_A^B \vec{F}_{21}\cdot d\vec{r}_1+\int_A^B\vec{F}_{12}\cdot d\vec{r}_2\tag{1}$$
$$=\ldots$$
$$=\int_A^B \vec{F}_{21}\cdot d\vec{r}_{21}\tag{2}$$
where ##A## and ##B## represent states A and B of the system.
(2) is be definition of the integral itself equal to ##\Delta K_{sys}##.
For a gravitational field, we end up with
$$W=Gm_1m_2\left ( \frac{1}{r_f}-\frac{1}{r_i} \right )$$
which shows that work depends only on the difference in initial distance ##r_i## and final distance ##r_f## between the two masses.
Change in potential energy of the system is defined as
$$\Delta U=-W=-\Delta K_{sys}$$
and by defining the potential energy at some reference point we obtain a potential energy function.
Now suppose our system is just object 1.
Object 2 creates a gravitational field and object 1 is in it. The force on object 1 is now an external force.
If object 2's position is fixed, then its gravitational field is also fixed at every point in space.
We can calculate the work done by this field on object 1 from an initial state to a final state and this will be path independent.
Thus, it seems we could define a potential energy function for object 1.
Of course, and here is what I am not really grasping very well, object 1 has a gravitational field of its own, and this affects object 2, which affects the latter's gravitational field if object 2's position changes due to acceleration.
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