Center of Mass and Tangential Gravitational Motion
by Ron Kurtus (updated 30 May 2023)
Tangential motion of two objects in space, with respect to the center of mass (CM) between them, is perpendicular to the radial line between the objects and through the CM. The objects move in opposite tangential directions with respect to the CM. While the radial motion components are a function of the gravitational force between the objects, tangential velocities are not affected by gravitation.
The tangential velocities determine whether the objects will collide, go into orbit or fly off into space. When there are additional radial velocities away from or toward the CM, the paths of the objects can be distorted.
Questions you may have include:
- How do objects move with respect to the CM?
- What effect does tangential motion have on the paths?
- What effect do radial velocities have on the paths?
This lesson will answer those questions. Useful tool: Units Conversion
Objects move in opposite directions
When the tangential components of the motion of two objects in space viewed with respect to the center of mass (CM), they are always in opposite directions, with the ratio of the velocities as:
mvTm = −MvTM
- m and M are the masses of the objects
- vTm and vTM are the tangential velocities of the objects
The negative sign indicates the velocity vectors are pointing in opposite directions. In other words, the objects are moving in opposite directions.
Factors in tangential motion
Ratio of speeds
Since speed is independent of direction:
msTm = MsTM
sTm/sTM = M/m
where sTm and sTM are the speeds or magnitudes of velocities vTm and vTM.
If the velocities do not appear to be in the given ratio or if both objects are moving in the same direction, the viewpoint is with respect to an outside observer and not relative to the CM. In these cases, though, the CM may appear to be moving in order to maintain the correct ratio.
(See Relative Motion and Center of Mass for more information.)
Results determined by velocities
Although the objects are moving perpendicular to the axis between them, they are still attracted toward each other by their gravitational force. The combination of the tangential velocities and the gravitational attraction toward the CM along the radial direction causes the objects to move in curved paths. The result of the curved motion is an outward centrifugal forces on each of the objects.
Possible paths of the objects are that they:
- Go into orbit
- Small elliptical orbits
- Circular orbits
- Large elliptical orbits
- Fly off into space
- Parabolic paths
- Hyperbolic paths
(See Effect of Velocity on Orbital Motion for more information.)
When the tangential velocities are zero (vTm = vTM = 0), the only motion is toward the CM, where the objects collide.
Go into small elliptical orbits
When the tangential velocities are less than required for a circular orbit, the objects will follow shallow elliptical paths. Depending on the physical size of the objects, they may either collide or go into orbit around each other.
Go into circular orbits
At specific tangential velocities for given masses and separations, the two objects will rotate about the CM between them in circular orbits. The centrifugal forces of the objects keep them in this stable orbit.
Note: Assume that there is no extra radial velocity, either toward or away from the CM that would affect the orbit.
The tangential velocities of the objects with respect to the CM are:
vTm = √[GM2/R(M + m)]
vTM = √[Gm2/R(M + m)]
- vTm and vTM are the tangential velocities
- G is the Universal Gravitational Constant
- M and m are the masses of the two objects
- R is the separation between the objects, as measured from their centers of mass
(See Derivation of Circular Orbits Around Center of Mass for more information.)
Go into elliptical orbits
When the velocities are greater than those required for circular orbits but less than the escape velocity, the objects will go into elliptical orbits around the CM. The velocity range for mass m is:
√[GM2/R(M + m)] < vTm < √[2GM2/R(M + m)]
Follow parabolic paths
If the velocities are at the escape velocity, the objects will take parabolic paths into space. The velocity for mass m is:
vTm = √[2GM2/R(M + m)]
If the velocity is with respect to the other object, the resulting escape velocity is:
vT = √[2G(M + m)/R]
(See Overview of Gravitational Escape Velocity for more information.)
Follow hyperbolic paths
If the velocities are above the escape velocity, the objects will follow hyperbolic paths and go off into space.
Effect of initial radial velocity
Besides the radial velocity from the gravitational attraction of the two objects, there may be an initial velocity, either toward the CM or away from the CM. In both cases, the paths of the objects are distorted according to the constant radial velocity.
Radial velocity toward CM
When the objects are moving in circular orbits, motion toward the CM is compensated by motion away from the centrifugal force. However, when the objects are moving toward the CM at some constant velocity, the effect would be to reduce the separation and change the shape of the paths of the objects.
The results could be changing the circular orbits into elliptical or even spirals into the CM.
Radial velocity away from CM
When the objects are moving away from the CM, such that they reach a maximum displacement before falling toward the CM, the effect of the tangential velocity is distorted.
For example, in order to be in a circular orbit, the tangential velocity must create a sufficient centrifugal force to equalize the gravitational force. However, the motion away from the CM increases the separation, thus requiring a higher tangential velocity for the orbits. The result would be elliptical shaped orbits until the objects returned to the initial separation needed for circular orbits. It is not easy to visualize, and the math required is complex.
Of course, if the outward radial velocities were at or above the escape velocities, the objects would follow curved paths as they flew into space.
Tangential motion of two objects in space is perpendicular to the radial line between the objects and in opposite directions with respect to the CM. Tangential velocities are not affected by gravitation.
The tangential velocities determine whether the objects will collide, go into orbit or fly off into space. When there are additional radial velocities, the paths of the objects are distorted.
Focus and don't go off on a tangent
Resources and references
Center of Mass Calculator - Univ. of Tennessee - Knoxville (Java applet)
Center of Mass - Wikipedia
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Center of Mass and Tangential Gravitational Motion