Two Bodies rotating around each other

[CA_Jones]

Hi I was wondering how the gravitational force of attraction is related to centripetal acceleration for two bodies rotating around a point between them.

Specifically how would you determine the centripetal acceleration of each body if their masses and the distance between them was given?

THanks

 

[Jolb]

To find the "centripetal" accelleration, find the center of mass and project the gravitational force vector onto the radius vector (from the center of mass).

I think your question might be better answered by something called the "two body problem." The way to solve for the trajectories of two orbiting bodies is by solving the orbit of something called the "reduced mass" around a force center located at the center of mass of the two bodies. You then use some equations to recover the trajectories of the two bodies from the reduced mass's trajectory.
http://en.wikipedia.org/wiki/Two-body_problem

 

[sganesh88]

The OP was asking about the centripetal acceleration only. Isn't it just equal to Gm1/r^2 and Gm2/r^2 respectively?
Where m1 and m2 are the masses; and r, the distance between them which he says are given.

 

[rcgldr]

Hi I was wondering how the gravitational force of attraction is related to centripetal acceleration for two bodies rotating around a point between them.

Gravitational force is a function of mass and distance, not velocity (ignoring the speed of gravity). The velocities determine the path of the objects, which was covered in the wiki link previously posted.

 

[Jolb]

The OP was asking about the centripetal acceleration only. Isn't it just equal to Gm1/r^2 and Gm2/r^2 respectively?
Where m1 and m2 are the masses; and r, the distance between them which he says are given.

Since two body motion is planar, we can use polar coordinates, r and theta. Closed Newtonian orbits are always ellipses (or circles, but that's just a special case of an elliptical orbit). Because the center of mass of the system is the only inertial thing in the problem, and since it's pretty common knowledge that objects orbit the center of mass, we can define the "centripetal" accelleration as the accelleration toward the center of mass. (In latin, "centri-" refers to the center and "petal" means "seeking", i.e., centripetal motion seeks the center, and the center of mass is our only viable "center" here.)

That's why I said you can just project the gravitational force onto the radius vector to find the centripetal force, then divide by mass to get the centripetal acceleration.

The reason that quantity is NOT equal to GMm/(r^2) is for this reason: bodies in (non-circular) elliptical orbits have a varying *angular* velocity, which means there must be some sort of angular acceleration/force. The only force on one body is the gravitational force from the other, and the other body's gravitational force must be supplying both the centripetal AND angular acceleration. Because the angular direction is always perpendicular to the direction to the center (the radial direction), any force keeping the body in a closed orbit and causing the body's angular velocity to increase must have both an angular component and a "centripetal" or radial component. The vector sum of angular force and centripetal force is equal to the total force on the body. Therefore the gravitational force is greater than or equal to the centripetal force.

 

[sganesh88]

Jolb- thanks a lot for the clarification. "
The component of the gravitational force contributing a torque about the COM cannot contribute to the centripetal acceleration too. I missed the point considering only circular orbit.