How can it be proved that every body has (and only one) CM point?

In summary, the conversation discusses the proof of the existence and uniqueness of the center of mass point for any given body. This is done by defining the center of mass point as the point where the mass-weighted relative positions sum up to 0, and showing that there cannot be any other points that satisfy this property. The conversation also mentions that this proof was required for a specific purpose, and the speaker was able to prove it despite not initially knowing the definition.
  • #1
Mad_Eye
69
0
how can it be proved that every body has (and only one) CM point?

given the center of the mass is the point that the distances relative to gives:
[tex]\sum m \cdot r = 0[/tex]
 
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  • #2
You define
[tex]r_c=\frac{\sum m_i r_i}{\sum m_i}[/tex]
Then you check
[tex]\sum m_i (r_i-r_c)=\sum m_i r_i-r_c \sum m_i=0[/tex]
 
  • #3
oh yeah... didn't know this is the definition...
was able to prov what i wanted nonetheless though :D
good thing i asked anyway :D
 
  • #4
That is totally wrong, Gerenuk!

SUPPOSE we define ONE C.M as
[tex]r^{(1}}_{C.M}=\frac{\sum_{i}m_{i}r_{i}}{M}[/tex]
It can then readily be shown that we have:
[tex]\frac{\sum_{i}m_{i}(r_{i}-r^{(1)}_{C.M})}{M}=0 (*)[/tex]
That is all you did, Gerenuk, NOW follows the proof you should have given:

Assume that there is another point, [itex]r^{(2)}[/tex] that satisfies (*) by taking the place of the defined C.M.

Then, we have:
[tex]r^{(1)}_{C.M}-r^{(2)}=\frac{\sum_{i}m_{i}r_{i}}{M}-\frac{M}{M}r^{(2)}=\frac{\sum_{i}m_{i}(r_{i}-r^{(2)})}{M}=0[/tex]
whereby uniqueness of C.M has been proven:
[tex]r^{(1)}_{C.M}-r^{(2)}=0[/tex]
 
  • #5
"Missing a step" doesn't equate to "totally wrong"
 
  • #6
diazona said:
"Missing a step" doesn't equate to "totally wrong"
Yes, it does, since that missing step was precisely the proof the asked for.
 
  • #7
yeah that what i did but..
if the definition of CM is Gerenuk's first equation, then there is no need to proof there is only one...
(though i did ask to use another given haha)
 
  • #8
Mad_Eye said:
yeah that what i did but..
if the definition of CM is Gerenuk's first equation, then there is no need to proof there is only one...
(though i did ask to use another given haha)
Not at all!

That definition proves that there exists AT LEAST 1 C.M.

It remains to prove that there are no other points having the same property (i.e, uniqueness of the point where the mass-weighted relative positions sums up to 0).

Another argument, ASSUMING the existence of (at least one) point satisfying your equation, can prove that IF such a point exists, then it must be unique.
(You'll need to CONSTRUCT such a point afterwards in order to prove that it does, indeed, exist!)

The uniqueness argument goes then as follows:

Assume that two such points exist. Then we have:
[tex]\sum_{i}m_{i}(r_{i}-r^{(1)})-\sum_{i}m_{i}(r_{i}-r^{(2)})=0[/tex]
since both terms are, by definition, equal to 0.
The right-hand side is now easily re-written as:
[tex](r^{(1)}-r^{(2)})\sum_{i}m_{i}=0[/tex]
and since the total mass is a positive number, uniqueness follows.
 

Related to How can it be proved that every body has (and only one) CM point?

1. How is the center of mass (CM) point defined in a body?

The center of mass point is defined as the point where the mass of a body is evenly distributed in all directions. It is also known as the point of balance or the point where the body would balance on a fulcrum.

2. Why is it important to prove that every body has a single CM point?

Proving that every body has a single CM point is important because it helps us understand the motion and stability of objects. It also allows us to accurately calculate the motion and forces acting on a body.

3. How can the CM point be determined experimentally?

The CM point can be determined experimentally by performing a balance test. This involves suspending the body from different points and finding the point where it balances. The point of balance is the CM point.

4. Can the CM point be outside the physical boundaries of a body?

No, the CM point cannot be outside the physical boundaries of a body. It must lie within the body, as it represents the average position of the body's mass.

5. Is the CM point the same for all reference frames?

Yes, the CM point is the same for all reference frames. This is because it represents the average position of the body's mass, which is independent of the observer's frame of reference.

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