Find Mass Center of Body Homework - 65 Characters

[xzibition8612]

Homework Statement

See attachment.

Homework Equations

The Attempt at a Solution

I don't get the first equation. Why is it 2pALx on the left side? Aren't we finding the mass center of the second shorter body? So shouldn't it then be (5/2)pALx? On the right side, I know pAL(L/2) is the mass center of the orange part on the left, but where did pAL(5L/4) come from? It seems like the 5L/4 is the mass center of the deformed total body, so I'm confused then why it seems to add over it twice. Any clear explanation would be appreciated, I'm obviously very confused about this.

 

[Filip Larsen]

For a composite object the center of mass $\overline{x}$ can be defined [1] as

$$m \overline{x} = \sum_i m_i x_i$$

where m is the total mass, and x_i and m_i is the CM position and mass of each component.

In the solution you refer to, the left and right hand side corresponds to the left and right side of the equation above, so you should be able to figure out each term by thinking about how to write up the total mass, the mass and CM position of the orange component and white component. For positions note that it is the position of the CM for that component relative to the common origin. In your case the origin is the left end of the orange component so the position of CM of the white component must "include" the length of the orange component.


[1] http://en.wikipedia.org/wiki/Center_of_mass

 

[xzibition8612]

I get it better now thanks. One more question, why is the left side 2pALx? I thought the question is asking about the deformed configuration, so shouldn't the left side be (5/2)pALx? Thanks again.

 

[Filip Larsen]

The mass on the left hand side refers to the total mass and since deformation does not change the total mass of the rod you can just as well calculate the mass from before deformation. If you were to calculate the mass after deformation (for instance by calculating the new density of the white component after deformation and sum up for the mass for the two components) you would end up with the same expression for the total mass.

 

[DeeNos]

The equation 2pALx on the left side represents the mass of the entire body, which includes both the longer and shorter bodies. This is because the mass center calculation takes into account the mass of the entire body, not just one section. The equation (5/2)pALx on the right side is the mass center of the shorter body, but it is being multiplied by (5L/4) because the shorter body is located 5L/4 away from the origin, as shown in the attached image. This is why it is added twice - once for the left side and once for the right side. I hope this clarifies the equation for you. Let me know if you have any other questions.