#21 回覆: COM
發表於 : 週日 5月 03, 2015 2:26 pm
The first version of my definition of COM : "In physics, the center of mass is a point in space where the weighted relative position of the distributed mass sums to zero."
So turned out it is too technical for someone to understand. I already answered it is a weighted function and in this "weight function" calculation, the influence is the mass. Again, that's still too technical. Okay, okay, let's put it in a even less technical way then.
If you glue two balls (ball A and ball B, each has perfectly even mass density) with same radius (radius = r) together and create one object. I call this object has 2 distributed mass located, each located at the center of ball A and B.
1) If the mass of ball A and B are exactly equal. (note: I didn't say "weight" the noun, I said "mass"). Now both ball has the equal amount of influence (same mass) in the calculation. The COM of this glued object is located exactly half way between the center of Ball A and Ball B. This is where the sum of "relative position of ball center A" and "relative position of ball center B" under equal influence (same mass) equal to zero.
2) if the mass of ball A is twice of ball B (note: I didn't say "weight" the noun, I said "mass"). In the same calculation of COM, ball A will have twice the influence compare to ball B (mass of A is 2 times of mass of B). The COM of this glued object is located in the line connecting ball center A and B, but closer to center of ball A. It is located at along the line jointing the center of ball A and B with 2/3 r (radius = r) away from ball A center. It is also 4/3 r away from ball B center on the exact opposite direction.
At this COM location:
1) relative position of ball A center is 2/3 r
2) relative position of ball B center is -4/3 r
3) influence of ball A is 2
4) influence of ball B is 1
"Weighted sum" of the relative position
= relative position of ball A * influence of ball A + relative position of ball B * influence of ball B
= 2/3 r * 2 + (-4/3 r) * 1
= 4/3 r - 4/3 r
= 0
[Nowhere above involves any gravity, force, etc. Just mass in space.]
[I didn't said anything like put the “weight” into the “mass”. I don't know how one could create that term and got confused. I cannot answer this one question at all.]
[I agree with “Arithmetic” is only dealing with the techniques of the calculation. Weighted sum, weighted average is just like +, -, *, /. It is a technique of calculation. There is nothing special about it and it is just another "arithmetic". Everything could make use of this kind of calculation. It is wrong for you to say it used by statistic alone. Physics calculation use all kinds of arithmetic in calculation. Weighted average for complex case in COM calculation, like uneven density or odd shaped object needs, to use integration. Is calculating an integral considered as a “math” by “pure” mathematician?]
[Again, I'm 100% sure what's I'm talking about here and that's the COM meaning used by most people in this world. If you don't like it, you are welcome to prove me wrong.]
[There is nothing wrong for me to say COM = Center of Gravity if it is under an uniform gravity field. Is there anything wrong to say the "length of a rod" is equal to "the height of the rod end when you place the other end vertically on the floor"? When there is no "floor" around, the second part of the measurement is no longer valid. However, that doesn't mean the rod has no length. The rod length doesn't change at all, whether or not there is a floor around. The same for COM, COM doesn't change location whether gravity is involved or not since gravity is not being involved in my definition of COM.]
So turned out it is too technical for someone to understand. I already answered it is a weighted function and in this "weight function" calculation, the influence is the mass. Again, that's still too technical. Okay, okay, let's put it in a even less technical way then.
If you glue two balls (ball A and ball B, each has perfectly even mass density) with same radius (radius = r) together and create one object. I call this object has 2 distributed mass located, each located at the center of ball A and B.
1) If the mass of ball A and B are exactly equal. (note: I didn't say "weight" the noun, I said "mass"). Now both ball has the equal amount of influence (same mass) in the calculation. The COM of this glued object is located exactly half way between the center of Ball A and Ball B. This is where the sum of "relative position of ball center A" and "relative position of ball center B" under equal influence (same mass) equal to zero.
2) if the mass of ball A is twice of ball B (note: I didn't say "weight" the noun, I said "mass"). In the same calculation of COM, ball A will have twice the influence compare to ball B (mass of A is 2 times of mass of B). The COM of this glued object is located in the line connecting ball center A and B, but closer to center of ball A. It is located at along the line jointing the center of ball A and B with 2/3 r (radius = r) away from ball A center. It is also 4/3 r away from ball B center on the exact opposite direction.
At this COM location:
1) relative position of ball A center is 2/3 r
2) relative position of ball B center is -4/3 r
3) influence of ball A is 2
4) influence of ball B is 1
"Weighted sum" of the relative position
= relative position of ball A * influence of ball A + relative position of ball B * influence of ball B
= 2/3 r * 2 + (-4/3 r) * 1
= 4/3 r - 4/3 r
= 0
[Nowhere above involves any gravity, force, etc. Just mass in space.]
[I didn't said anything like put the “weight” into the “mass”. I don't know how one could create that term and got confused. I cannot answer this one question at all.]
[I agree with “Arithmetic” is only dealing with the techniques of the calculation. Weighted sum, weighted average is just like +, -, *, /. It is a technique of calculation. There is nothing special about it and it is just another "arithmetic". Everything could make use of this kind of calculation. It is wrong for you to say it used by statistic alone. Physics calculation use all kinds of arithmetic in calculation. Weighted average for complex case in COM calculation, like uneven density or odd shaped object needs, to use integration. Is calculating an integral considered as a “math” by “pure” mathematician?]
[Again, I'm 100% sure what's I'm talking about here and that's the COM meaning used by most people in this world. If you don't like it, you are welcome to prove me wrong.]
[There is nothing wrong for me to say COM = Center of Gravity if it is under an uniform gravity field. Is there anything wrong to say the "length of a rod" is equal to "the height of the rod end when you place the other end vertically on the floor"? When there is no "floor" around, the second part of the measurement is no longer valid. However, that doesn't mean the rod has no length. The rod length doesn't change at all, whether or not there is a floor around. The same for COM, COM doesn't change location whether gravity is involved or not since gravity is not being involved in my definition of COM.]