Two Bodies rotating around each other

In summary: The centripetal acceleration is the acceleration towards the center of mass due to the gravitational force. In summary, the gravitational force of attraction is related to centripetal acceleration for two rotating bodies. The centripetal acceleration is determined by the centripetal force and the mass of the bodies.
  • #1
CA_Jones
1
0
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
 
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  • #2
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
 
  • #3
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.
 
  • #4
CA_Jones said:
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.
 
  • #5
sganesh88 said:
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.
 
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  • #6
Jolb- thanks a lot for the clarification. :smile:"
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.
 

Related to Two Bodies rotating around each other

What is meant by "Two Bodies rotating around each other"?

"Two Bodies rotating around each other" refers to a system where two objects are orbiting each other due to their mutual gravitational attraction. This is also known as a binary system.

What are some examples of "Two Bodies rotating around each other" in the universe?

Some examples of "Two Bodies rotating around each other" include the Earth and Moon, the Sun and Earth, and the Milky Way galaxy and its satellite galaxies.

How do the masses and distances between the two bodies affect their rotation?

The masses and distances between the two bodies determine the strength of their gravitational attraction, which in turn affects the speed and shape of their orbits. Larger masses and closer distances result in stronger gravitational forces and faster orbital speeds.

What are some real-life applications of studying "Two Bodies rotating around each other"?

Studying "Two Bodies rotating around each other" can help us understand the dynamics of celestial bodies, such as predicting eclipses and tides. It also has practical applications in space exploration and satellite navigation.

How does the concept of "Two Bodies rotating around each other" relate to Newton's laws of motion?

The concept of "Two Bodies rotating around each other" is governed by Newton's laws of motion, specifically the law of universal gravitation. This law states that every object attracts every other object with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them.

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