Gravitation with multiple objects by using center of mass

In summary, the conversation discusses whether using center of mass is a valid method for calculating the resultant gravitational force on a sphere. It is argued that while this method may work for planets and large objects, it does not make sense as a general rule due to the fact that the gravitational force can become arbitrarily large when using center of mass. The recommended approach is to calculate the force of gravity from each individual sphere and add the vectors.
  • #1
XSethX
5
0
Four 9.5 kg spheres are located at the corners of a square of side .60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphere by the other three.

So, essentially my question is, why can I not use center of mass of the other 3 spheres to calculate the gravitational force? I set the top left sphere as my frame of reference, and found the center of mass of the bottom left and top right spheres and calculated the gravitational force at that center of mass on the top left sphere (I narrowed down my error to this part.) Why can I not do this? Thanks in advance!

Edit: Very very sorry, realized I should have posted this in "Introductory Physics Homework"...
 
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  • #2
I'm not sure exactly why you can't do it, but I also don't see why you would want to do it. I would calculate the force of gravity from each sphere and add the vectors.
 
  • #3
By the way, welcome to PF!
 
  • #4
XSethX said:
I set the top left sphere as my frame of reference, and found the center of mass of the bottom left and top right spheres and calculated the gravitational force at that center of mass on the top left sphere (I narrowed down my error to this part.)

What happened to the bottom right sphere?
 
  • #5
It won't generally work out if you use center of mass to calculate the resultant gravitational force.
(It is fortunate that it turns out true for planets and such.)
To see why it doesn't make sense as a general rule, consider the following question:
What is the gravitational force at the origin from two identical particles whose positions along some axis are -L+dL and L+dL? We expect (by symmetry) that as we choose smaller dL, the gravitational force should go to zero. Notice though, that the center of mass is a distance dL from the origin; so using your method the gravitational force gets arbitrarily large (we are dividing by dL squared). Clearly incorrect.
 
  • #6
Drakkith said:
What happened to the bottom right sphere?
I didn't forget, I just calculated that after and added to the number I found. Both times I tried to solve the problem, that value was the same, so I figured my error was with using the center of mass for the bottom left and top right spheres. Thanks!
 
  • #7
Isaac0427 said:
I'm not sure exactly why you can't do it, but I also don't see why you would want to do it. I would calculate the force of gravity from each sphere and add the vectors.
Yeah, that's how I got the right answer. I was just curious about why using center of mass didn't work. Thanks, though!
 
  • #8
Nathanael said:
It won't generally work out if you use center of mass to calculate the resultant gravitational force.
(It is fortunate that it turns out true for planets and such.)
To see why it doesn't make sense as a general rule, consider the following question:
What is the gravitational force at the origin from two identical particles whose positions along some axis are -L+dL and L+dL? We expect (by symmetry) that as we choose smaller dL, the gravitational force should go to zero. Notice though, that the center of mass is a distance dL from the origin; so using your method the gravitational force gets arbitrarily large (we are dividing by dL squared). Clearly incorrect.
Oh, I understand now. Thank you!
 

1. How is the center of mass calculated for multiple objects?

The center of mass for multiple objects is calculated by taking the sum of the individual masses multiplied by their respective positions, and then dividing by the total mass of all the objects. This can be represented by the equation: center of mass = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn), where m represents mass and x represents position.

2. How does the center of mass affect the motion of multiple objects?

The center of mass is the point where the total mass of an object can be considered to be concentrated. In a system with multiple objects, the center of mass will move according to the combined forces acting on all the objects. This means that the motion of the center of mass can be used to predict the overall motion of the system.

3. What is the relationship between the center of mass and gravitational forces?

The center of mass is the point where the gravitational forces of all the objects in a system balance out. This means that if an external force is applied to the system, the center of mass will move in the direction of the force. Additionally, the gravitational forces between the objects in the system can be simplified and treated as if they are acting on a single object at the center of mass.

4. Is the center of mass always located within the objects in a system?

No, the center of mass can be located outside of the objects in a system, depending on their distribution of mass. For example, in a system with two objects of equal mass, the center of mass will be located exactly halfway between the two objects. In a system with unequal masses, the center of mass will be closer to the heavier object.

5. How does the center of mass affect the stability of a system?

The location of the center of mass can affect the stability of a system. If the center of mass is located within the base of support of a system, it will be stable. However, if the center of mass is located outside of the base of support, the system will be unstable and may topple over. This is why it is important to consider the center of mass when designing structures or vehicles.

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