# Potential energy of a two-body system

• AlonsoMcLaren
In summary, the potential energy of a system consisting of two masses, m and 2m, separated by a distance r is given by U = - \frac{G m_1 m_2}{r}, where G is the gravitational constant. This potential energy is a property of the system and not of the individual masses. To derive it from the gravitational force, we can integrate along the radial and take the potential at infinity to be zero. If the masses are comparable, the center-of-mass coordinates and reduced mass must be used. However, for two bodies, either mass can be used as an inertial frame of reference. The same gravitational potential is obtained regardless.
AlonsoMcLaren
If two masses, m and 2m, are separated by a distance r, what is the potential energy of mass m? What is the potential energy of mass 2m? What is the potential energy of the system?

The gravitational PE is a property of the system, not of the individual masses.

Doc Al said:
The gravitational PE is a property of the system, not of the individual masses.

So what is the value of PE of the system?

AlonsoMcLaren said:
So what is the value of PE of the system?
The gravitational PE of two masses is given by:
$$U = - \frac{G m_1 m_2}{r}$$
Where m1 and m2 are the masses and r the distance between them.

Doc Al said:
The gravitational PE of two masses is given by:
$$U = - \frac{G m_1 m_2}{r}$$
Where m1 and m2 are the masses and r the distance between them.

How to derive it from F=Gm1m2/(r^2)

$$\Delta U = \int_{r_1}^{r_2}\vec{F}\cdot d\vec{r}$$

In this case, it's easiest to integrate along the radial, and taking potential at infinity to be zero, you get this.

$$U = \int_{\infty}^{r}G\frac{m_1 m_2}{r^2}dr = -G\frac{m_1 m_2}{r}$$

AlonsoMcLaren said:
How to derive it from F=Gm1m2/(r^2)
In general:
$$F = - \frac{dU}{dr}$$
So, to go from F to U, integrate.

(Oops... K^2 beat me to it.)

K^2 said:
$$\Delta U = \int_{r_1}^{r_2}\vec{F}\cdot d\vec{r}$$

In this case, it's easiest to integrate along the radial, and taking potential at infinity to be zero, you get this.

$$U = \int_{\infty}^{r}G\frac{m_1 m_2}{r^2}dr = -G\frac{m_1 m_2}{r}$$

So are you using m1 (or m2) as frame of reference? If so, the frame is non-inertial.

ilyandm
Depends. If m1>>m2, then the acceleration of m1 is negligible, and we can use it as an inertial frame of reference. Or vice versa. But if m1 and m2 are comparable, you are right, and we have to use center-of-mass coordinates. In that case, r is the distance from center of mass, and either m1 or m2 is the reduced mass.

The idea behind reduced mass is that with central potential, which gravity happens to be, instead of looking at two bodies orbiting each other, you can consider bodies one at a time, and treat them as if each orbits the center-of-mass point as if it was immovable point with gravitational attraction. The strength of attraction is determined by reduced mass, which you compute to be sufficient to generate same force as in the original setup.

It's a bit messy, but for two bodies it works fine. Throw in a third body, and it goes from messy to near-impossible.

P.S. Gravitational potential works out to be exactly the same, by the way, so it doesn't matter.

ilyandm

## 1. What is potential energy of a two-body system?

Potential energy of a two-body system is the energy that is stored in the system due to the relative position and arrangement of the two bodies.

## 2. How is the potential energy of a two-body system calculated?

The potential energy of a two-body system can be calculated using the equation: U = -G(m1m2)/r, where G is the gravitational constant, m1 and m2 are the masses of the two bodies, and r is the distance between them.

## 3. What factors affect the potential energy of a two-body system?

The potential energy of a two-body system is affected by the masses of the two bodies, the distance between them, and the gravitational constant.

## 4. Does the potential energy of a two-body system change as the bodies move?

Yes, the potential energy of a two-body system changes as the bodies move closer or further apart. It is highest when the bodies are farthest apart and decreases as they move closer together.

## 5. What is the relationship between potential energy and kinetic energy in a two-body system?

In a two-body system, potential energy and kinetic energy are directly related. As potential energy decreases, kinetic energy increases and vice versa. This is known as the conservation of energy.

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