Name: center of mass in the two-body problem
Uploaded: 2025-05-22T09:06:09.000Z
Duration: 1 min 12 s
Channel: JoVE

K12 Standards

Virginia  Standards of Learning

HIgh School

Earth Science

ES.3

ES.3.b

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Reduced Mass Coordinates: Isolated Two-body Problem

The earth rotates around the sun. According to Newton&#39;s law of gravitation, the sun and earth exert equal and opposite forces on each other. So why does only the earth revolve?
Consider two bodies moving at different velocities. The velocity of the center of mass of this system lies on the line joining their individual velocities.
If the system is isolated, the center of mass can be taken as the origin. The velocities of the bodies can now be expressed relative to the velocity of the center of mass.
Multiplying the velocities with their respective masses gives their momenta. The first term here denotes the reduced mass of the two-body system.
The equation of the two-body system is now converted into a one-body system.
In the earth-sun system, the center of mass velocity can be approximated to the velocity of the sun. So, the sun has negligible relative velocity with respect to its center of mass, whereas the earth possesses a relative velocity which allows it to revolve around the sun.

reduced mass coordinates: isolated two-body problem

In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an infinite mass. The reduced mass of such a system is given by
<img alt="Gravitational physics concept; reduced mass formula &#956;=m&#8321;m&#8322;/(m&#8321;+m&#8322;); mathematical equation." src="/files/ftp_upload/14740/14740_Equation_1.svg" style="text-align: center; height: 32px;" data-altupdatedat="1754904545000" data-alt="Effective mass formula, &#956;=(m1*m2)/(m1+m2); physics equation diagram." />
Consider a two-particle system of different masses at distances r1 and r2 from the origin. The center of mass of this system lies on the line joining the two particles and can be expressed as
<img alt="Center of mass formula, vector equation; physics concept, mass distribution, position vectors." src="/files/ftp_upload/14740/14740_Equation_2.svg" style="text-align: center; height: 40px;" data-altupdatedat="1754904545000" data-alt="Center of mass formula, vector diagram, Rcm = (m1r1 + m2r2) / (m1 + m2), physics concept." />
If the center of mass is considered the origin, it divides the line joining the two particles in the inverse ratio of their masses. When no external force is acting on the system of the two particles, and the only forces are mutual interactions, the system behaves as a single particle with a reduced mass. It acts at the center of the mass of the two particles. The importance of the center of mass and reduced mass lies in the fact that, instead of having two separate equations of motion, a single equation of motion involving reduced mass reduces a two-body problem to a one-body problem. In conclusion, reduced mass coordinates are a powerful tool in studying the two-body problem, particularly in the isolated two-body problem.

In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other ...

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center of mass in the two-body problem

Reduced mass coordinates help simplify the classical two-body problem. In classical mechanics, this problem describes the motion of two interacting bodies under gravity or another central force. The key idea is that the system can be treated like a single-body problem when reduced mass is used.

For this description, one body is imagined to move with the reduced mass while the other body is fixed in place with infinite mass. The reduced mass is a property of the two-body system and is defined from the masses of the two particles. This approach changes a pair of motions into one simpler motion.

The center of mass is also central to the analysis. For two particles with different masses at distances r1 and r2 from the origin, the center of mass lies on the line joining them. If the center of mass is chosen as the origin, it divides that line in the inverse ratio of the two masses.

When no external force acts on the two-particle system, the only forces are mutual interactions. In that case, the system behaves like a single particle with reduced mass located at the center of mass. Instead of two separate equations of motion, one equation is enough. That is why reduced mass coordinates are useful for the isolated two-body problem.

Center of Mass in the Two-Body Problem

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Reduced mass coordinates help simplify the classical two-body problem. In classical mechanics, this problem describes the motion of two interacting bodies under gravity or another central force. The key idea is that the system can be treated like a single-body problem when reduced mass is used. For this description, one body is imagined to move with the reduced mass while the other body is fixed in place with infinite mass. The reduced mass is a property of the two-body system and is defined...

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key terms and concepts

Learn how reduced mass coordinates and the center of mass simplify the isolated two-body problem by turning two interacting bodies into one equation of motion.

Center of Mass in the Two-Body Problem | JoVE K12

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Center of Mass in the Two-Body Problem

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