Conceptual/mathematical center of mass question

In summary, the center of mass can be expressed in sigma notation as xcom = 1/M * SIGMA mixi, where M is the total mass and mixi represents the weighted average of the x-values. This can also be expressed in integral form as xcom = 1/M * INT xdm, where "dm" represents a tiny element of mass and "x" is the position of that element. While integrating with respect to "m" may suggest that x is dependent on m, it is simply a way to break down the total mass into smaller elements. Using "mdx" in the integral does not make sense as "m" would be the mass of what and "dx" would not represent the position
  • #1
quincyboy7
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The sigma notation of the center of mass is

xcom = 1/M * SIGMA mixi. I understand this because you are just taking a "weighted average" of sorts to find the correct x-value. My difficulty in understanding arises when this is expressed in integral form i.e.

xcom=1/M * INT xdm. First of all, doesn't integrating with respect to m suggest that x is dependent on m? Similarly, what does a "dm" mean in terms of an x-value? Couldn't an infinitesimal amount of mass exist anywhere on the x-axis? Wouldn't mdx make a lot more sense in the integral? Any help would be appreciated.
 
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  • #2
quincyboy7 said:
xcom=1/M * INT xdm. First of all, doesn't integrating with respect to m suggest that x is dependent on m?
You'll need to express "dm" in terms of "dx" in order to integrate.
Similarly, what does a "dm" mean in terms of an x-value?
You are taking a tiny element of mass "dm" and multiplying it by its x-value. This is exactly the same thing that you did in the first formula (with SIGMA instead of INT), except there the pieces were macroscopic with mass "m" instead of "dm".
Couldn't an infinitesimal amount of mass exist anywhere on the x-axis?
Of course.
Wouldn't mdx make a lot more sense in the integral?
No. What would "m" be the mass of? The point with the integral is that the mass is now continuously distributed, so we have to break the total mass into a zillion tiny elements (dm) and integrate. And what would "dx" mean in that expression? We want the position of each tiny piece of mass, which is x, not dx.
 

1. What is the conceptual center of mass?

The conceptual center of mass is a point that represents the average position of the mass of an object. It takes into account both the mass and the distribution of mass within the object.

2. How is the mathematical center of mass calculated?

The mathematical center of mass is calculated by finding the weighted average of the individual masses within an object. This is done by multiplying the mass of each component by its distance from a chosen reference point and then dividing the sum of these products by the total mass.

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

The center of mass is important in physics because it is the point at which an object can be balanced. It also helps in analyzing the motion of an object as it is the point at which the net force acts and determines the direction of motion.

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

Yes, the center of mass can be located outside of an object if the mass distribution within the object is uneven. For example, a hammer has its center of mass located closer to the head because the head has more mass compared to the handle.

5. How does the center of mass change when an object is in motion?

The center of mass remains the same in both motion and rest. However, when an object is in motion, its center of mass may undergo displacement or acceleration depending on the external forces acting on it.

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