Potential energy of a system of particles

[Rick16]

TL;DR

I have a conceptual problem with the potential energy of a system of particles

Here is problem 1.55 (a) from Schroeder's Thermal Physics:

Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is -2 times the total kinetic energy.

I did not know how to find the potential energy of this system, so I looked up the solution, according to which the potential energy of the system is simply the potential energy at the location of one of the particles. Why would that be? And what if I have a system with more than 2 particles? Would the potential energy of the system as a whole be the same as the potential energy at the position of one of the particles due to the presence of the other particles?

 

[A.T.]

I looked up the solution, according to which the potential energy of the system is simply the potential energy at the location of one of the particles

Can you post the exact wording of the solution?

 

[Rick16]

Can you post the exact wording of the solution?

Here is the solution:

The kinetic and potential energies of the system are $$U_k=2\cdot\frac{1}{2}mv^2=mv^2,~~~U_p=-\frac{Gm^2}{2r}.$$ To show how these are related, apply Newton's second law to the motion of one of the particles: $$F=ma\Rightarrow \frac{Gm^2}{(2r)^2}=m\frac{v^2}{r}.$$ Multiply each side of this equation by $2r$, and the left-hand side is the magnitude of the potential energy. Therefore, $$U_p=-2mv^2=-2U_k$$.

 

[Rick16]

One more comment to show where exactly my problem lies: Potential energy is defined at a specific position. A System does not have a specific position. How then can I define/understand what the potential energy of a system would be?

 

[PeroK]

One more comment to show where exactly my problem lies: Potential energy is defined at a specific position. A System does not have a specific position. How then can I define/understand what the potential energy of a system would be?

Potential energy is generally a function of the position of all particles in the system.

 

[PeroK]

TL;DR: I have a conceptual problem with the potential energy of a system of particles

Here is problem 1.55 (a) from Schroeder's Thermal Physics:

Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is -2 times the total kinetic energy.

I did not know how to find the potential energy of this system, so I looked up the solution, according to which the potential energy of the system is simply the potential energy at the location of one of the particles. Why would that be? And what if I have a system with more than 2 particles? Would the potential energy of the system as a whole be the same as the potential energy at the position of one of the particles due to the presence of the other particles?

It's a good idea to work this out for yourself. Take two particles of different masses at rest relative to each other. Release them and calculate the total KE upon impact.

 

[Herman Trivilino]

By convention we usually assume that as the separation distance increases beyond all bounds, the potential energy approaches zero. That's a mouthful so we just say "the potential energy is zero at infinity".

, according to which the potential energy of the system is simply the potential energy at the location of one of the particles. Why would that be?

Following the above convention, the potential energy of a system is the energy it takes to assemble the constituents of the system.

Thus if you have only one constituent in otherwise empty space, it takes no energy to move the first object into position.

Then the energy to move the second object into position is the total energy to assemble the constituents of the system.

And what if I have a system with more than 2 particles?

Again, it's the energy it takes to assemble the system.

 

[Rick16]

By convention we usually assume that as the separation distance increases beyond all bounds, the potential energy approaches zero. That's a mouthful so we just say "the potential energy is zero at infinity".



Following the above convention, the potential energy of a system is the energy it takes to assemble the constituents of the system.

Thus if you have only one constituent in otherwise empty space, it takes no energy to move the first object into position.

Then the energy to move the second object into position is the total energy to assemble the constituents of the system.



Again, it's the energy it takes to assemble the system.

Thank you. This is it. It is not even new to me. I read about it probably more than once, but I had completely forgotten about it. This shows what happens when you just read physics texts without doing problems.