Potential energy of a system of particles

In summary, the participants in this conversation discuss the concept of total potential energy in a system of two particles and whether it is equal to the potential energy of one particle with the other fixed. They consider the general case and use Newton's laws and the concept of relative velocity to prove that the total kinetic energy of the system is conserved. They also mention the importance of the potential being a function of the relative position in order for this proof to hold.
  • #1
quasar987
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Hi.

In trying to convince myself that the total potential energy of a system of two particles was just the potential energy of one particle computed while considering the other particle fixed, I imagined two identical particles a distance 2r apart and attracting each other, and concluded that their total kinetic energy when they meet would be the same if both particles were free as if one was fixed.

But I don't know how to prove the thing in the most general case possible. Can anyone help?

Thanks.
 
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  • #2
Sure enough:
This is because the interaction potential must depend (as in the cases of gravitational/electrostatic potentials) on the particles RELATIVE position.
Let us state Newton's 2.laws on 2 particles, along with Newton's 3.law:
[tex]\vec{F}_{12}=m_{1}\vec{a}_{1} (1)[/tex]
[tex]\vec{F}_{21}=m_{2}\vec{a}_{2}(2)[/tex]
[tex]\vec{F}_{12}=-\vec{F}_{21}[/tex]
(I hope the notations speak for themselves..)
Now, form the dot products with the respective velocities, and add them together:
[tex]\vec{F}_{12}\cdot\vec{v}_{1}+\vec{F}_{21}\cdot\vec{v}_{2}=\frac{d}{dt}(\frac{1}{2}m_{1}\vec{v}_{1}^{2}+\frac{1}{2}m_{2}\vec{v}_{2}^{2})[/tex]
Or, identifying the kinetic energy of the system [tex]\mathcal{K}[/tex] along with use of 3.law:
[tex]\vec{F}_{12}\cdot\vec{v}_{12}=\frac{d\mathcal{K}}{dt}(3)[/tex]
where I have introduced the relative velocity of particle 1, with respect to particle 2.

Now, let [tex]\vec{F}_{12}=-\nabla{V}_{12}, V_{12}=V_{12}(\vec{r}_{12})[/tex]
that is, the potential is a function of the relative velocity.
This is required in order that we may integrate (3) with respect to time to gain conservation of mechanical energy:
[tex]\mathcal{K}+V_{12}=C[/tex]
 
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Likes ilyandm
  • #3
Fantastic! Thanks arildno!
 

What is potential energy?

Potential energy is the energy possessed by an object due to its position or configuration. It is the energy that is stored in a system and can be converted into other forms of energy.

How is potential energy of a system of particles calculated?

The potential energy of a system of particles is calculated by summing up the potential energy of each individual particle. The formula for potential energy is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object from a reference point.

What factors affect the potential energy of a system of particles?

The potential energy of a system of particles is affected by the mass of the particles, the distance between them, and the strength of the force acting between them. The potential energy also depends on the reference point chosen for calculation.

Can potential energy be negative?

Yes, potential energy can be negative. This occurs when the reference point for calculating potential energy is chosen at a higher level than the position of the object. In this case, the potential energy is lower than the reference point and is considered negative.

How can potential energy be converted into other forms of energy?

Potential energy can be converted into other forms of energy such as kinetic energy, thermal energy, or electrical energy. This can happen when the object moves and its potential energy is converted into kinetic energy, or when the object is released and its potential energy is converted into thermal energy due to friction. Potential energy can also be converted into electrical energy through devices like generators.

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