Intuition behind Centroids / Center of Mass

[paulclou]

Hi all,

I've been digging around in Google as well as searching under physics forum for a while now but I still can't find the answer to my question. If there is already an answer then I'm sorry for wasting the server bandwith and database harddrive usage

I learned the formulation for center of mass/centroid a while ago but now it's popping up again and I'm wondering why, do we need to multiply x and y with dA or dV, then divide by dA or dV? I understand it's trying to find the average location of the center of mass or centroid but why, just by multiplying dA with x, then divide by dA will get you the location of the centroid / center of mass? And doesn't multiplying dA with x goes to the third dimension?

Perhaps, a better question to ask is, what is the purpose of intergal(x*dA)?
Thanks,
Paul

 

[Hassan2]

Hi all,

I've been digging around in Google as well as searching under physics forum for a while now but I still can't find the answer to my question. If there is already an answer then I'm sorry for wasting the server bandwith and database harddrive usage

I learned the formulation for center of mass/centroid a while ago but now it's popping up again and I'm wondering why, do we need to multiply x and y with dA or dV, then divide by dA or dV? I understand it's trying to find the average location of the center of mass or centroid but why, just by multiplying dA with x, then divide by dA will get you the location of the centroid / center of mass? And doesn't multiplying dA with x goes to the third dimension?

Perhaps, a better question to ask is, what is the purpose of intergal(x*dA)?
Thanks,
Paul

According to the difinition M\bar{x}=∫xdm, but we have dm=\rhodv where \rho is the mass density.

Therefore \bar{x}=\frac{∫ρxdv}{M}

in the above equations, M is the total mass which is equal to ∫ρdv.

Therefore, \bar{x}=\frac{∫xρdv}{∫ρdv}.

When ρ is constant, \bar{x}=\frac{∫xdv}{V}, where V is the total volume.

If you want to know about the meaning of the definition, it's like finding a point where the moment due to total mass on that point is equal to the sum of the moments due to the distributed mass.

I hope that helps.