Center of mass and distance ratio

In summary, to show the inverse ratio of distance between two particles from their centers of mass, one can use the definition of CM and assume that the coordinate of the CM is zero. This can be achieved by a change in the position of the origin of x. The ratio of distances to the CM can then be computed, taking into account that one distance will be negative.
  • #1
kthouz
193
0
how to show that the ratio of the distance between two particles from their centers of mass is the inverse of their masses? I tried but i found a way which can be possible if only we assume that they are constituted by a same number of small particles.
 
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  • #2
Start with the definition of CM:
[tex] x_{cm}={m_1\ell_1+ m_2\ell_2 \over m_1+m_2}[/tex]
Then ask that the coordinate of the CM be zero. This is just a change in the position of the origin of x.
Then compute de ratio of distances to the CM. Don't forget that now one of the distances in negative. See why?
 
  • #3


I can confirm that the ratio of the distance between two particles from their centers of mass is indeed inversely proportional to their masses. This is known as the center of mass and distance ratio, and it is a fundamental concept in physics.

To understand this, we must first define what the center of mass is. The center of mass is the point at which the mass of a system can be considered to be concentrated. In other words, it is the average position of all the mass in a given system. For a system of two particles, the center of mass is located at a point between the two particles, and its position is determined by the masses and distances of the particles.

Now, let's consider two particles with different masses, M1 and M2, located at a distance r from each other. The ratio of the distance between their centers of mass, d, to their masses is given by the equation d/M1 = (r-d)/M2. This can also be written as d/M1 + d/M2 = r. By rearranging this equation, we can see that d/M1 is equal to r-d/M2. This means that the ratio of the distance between the centers of mass is inversely proportional to the masses of the particles, as stated in the question.

To show this mathematically, we can use the concept of the center of mass formula, which states that the center of mass of a system is equal to the sum of the products of each mass and its respective distance from a chosen reference point, divided by the total mass of the system. In this case, our reference point is the center of mass of the entire system.

Using this formula, we can calculate the center of mass for our two-particle system as (M1d1 + M2d2)/(M1+M2), where d1 and d2 are the distances from the center of mass to particles 1 and 2, respectively. We can also express this as (M1d1 + M2(r-d1))/(M1+M2). By setting this equal to zero (since the center of mass is located at a point where the net force on the system is zero), we can solve for d1 and get d1 = r(M2/(M1+M2)). Similarly, we can solve for d2 and get d2 = r(M1/(M1+M2)).

Now, if
 

1. What is the center of mass?

The center of mass is the point at which the mass of an object is evenly distributed, or the point where the object will balance if suspended.

2. How is the center of mass calculated?

The center of mass is calculated by taking the weighted average of the positions of all the particles that make up an object. This can be done using the formula: xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn), where xcm is the center of mass and m1, m2, etc. are the masses of the particles.

3. What is the significance of the center of mass?

The center of mass is important in understanding the stability and motion of objects. It is also used in various physics calculations, such as determining the torque on an object.

4. How does the distance from the center of mass affect an object's motion?

The distance from the center of mass affects an object's rotational motion. The farther an object's mass is from its center of mass, the greater the torque will be on the object, causing it to rotate more easily.

5. Can the center of mass be located outside of an object?

Yes, the center of mass can be located outside of an object. This can happen if the object has an irregular shape or if the mass is not evenly distributed. In these cases, the center of mass may be located at a point outside of the physical boundaries of the object.

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