Position of CM for 3 Cubes Along a Line

In summary, the conversation discusses the calculation of the center of mass of three cubes placed in contact with each other, with the l=2l cube in the center. The equation for center of mass is given, and the total mass of the cubes can be found by calculating the density and volume. A diagram is suggested to aid understanding, and the final answer for the center of mass is determined to be 3.83L. The question of whether the center of mass would be outside of the object is addressed and clarified.
  • #1
vinny380
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Question: Three cubes of sides l, 2l, and 3l are placed next to one another (in contact) with their centers along a straight line and the l=2l cube in the center. What is the position, along the line, of the CM of this system? Assume the cubes are made of the same uniform material.

My reasoning: So the equation for CM= M1XI +M2X2 + M3X3/ TM ... where M= mass, and X= distance ...So M(l +2l +3l)/3m = CM
M(6L)/3m = 2L
So, I got the center of Mass is 2L

I don't think my answer is correct, and even if my approach is correct. It is labeled a pretty easy problem, but I get really confused with problems without numbers. Also, I am not sure if you can conclude the total mass is 3M considering that means all the cubes would have to be the same mass ... and I also do not know what the question maker meant when he wrote the l=2l cube in the center ...

Please help!
 
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  • #2
The coordinates X1, X2, and X3 are the coordinates of the center of mass of each cube. So, all you have to do is place the origin wherever you want and start to calculate. If you place the origin at the beginning of the first cube, then X1 = 0.5 L, and so on..
 
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  • #3
radou ... thanks for the help but i am still confused... how would you find the total mass of the cubes??
 
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  • #4
vinny380 said:
radou ... thanks for the help but i am still confused... how would you find the total mass of the cubes??

Well, since the cubes are made of the same uniform material, you may assume the density of the cubes is equal. You know the volume, so, you can calculate the mass.
 
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  • #5
finding the volume is easy , but how would you go about finding the density? is it simply a known value?
 
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  • #6
vinny380 said:
finding the volume is easy , but how would you go about finding the density? is it simply a known value?

Yes, call it [tex]\rho[/tex] or something. It will cancel out in the further calculation.
 
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  • #7
A diagram might help.

center-of-mass.png
 
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  • #8
is the answer 3.08Lo (thats what i got) ?
 
Last edited:
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  • #9
According to my calculation, it's 3.83L. But I may be wrong. Nevertheless, it's important you understand the principle. :smile:
 
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  • #10
yeaaa... i just did it again and got 3.83L ... but radou, how does that make sense??
 
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  • #11
vinny380 said:
yeaaa... i just did it again and got 3.83L ... but radou, how does that make sense??

What exactly do you mean?
 
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  • #12
well...if the center of mass is 3.83L ... then wouldn't the center of mass be out of the object given (which is impossible)?
 
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  • #13
vinny380 said:
well...if the center of mass is 3.83L ... then wouldn't the center of mass be out of the object given (which is impossible)?

No it wouldn't, because the total length of the object is L + 2L + 3L = 6L.
 
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  • #14
thanks radou!
 
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Related to Position of CM for 3 Cubes Along a Line

1. What is the position of the center of mass (CM) for 3 cubes placed along a line?

The position of the center of mass for 3 cubes placed along a line is the point where all the mass of the cubes is evenly distributed. This point can be found by taking the average of the positions of the individual centers of mass of each cube.

2. How is the position of the center of mass calculated for 3 cubes placed along a line?

The position of the center of mass for 3 cubes placed along a line can be calculated using the formula: x_cm = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3), where x_cm is the position of the center of mass, m1, m2, and m3 are the masses of each cube, and x1, x2, and x3 are the positions of the individual centers of mass of each cube.

3. Is the position of the center of mass affected by the size or shape of the cubes?

Yes, the position of the center of mass for 3 cubes placed along a line is affected by the size and shape of the cubes. The position of the center of mass is determined by the distribution of mass in the cubes, so any changes in the size or shape of the cubes will affect the position of the center of mass.

4. What is the significance of determining the position of the center of mass for 3 cubes placed along a line?

Determining the position of the center of mass is important in understanding the overall stability and balance of the system. It also helps in predicting the motion of the cubes under external forces.

5. Can the position of the center of mass for 3 cubes placed along a line change?

Yes, the position of the center of mass can change if the mass or positions of the individual cubes change. Additionally, if external forces are applied to the system, the position of the center of mass may also change.

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