Why Does Torque Calculation Assume Mass Concentrated at the Center of Mass?

[xailer]

hello

I'm learning about torques and got really stuck. I hope someone can take the time and help me out with this

When calculating the torque caused by the weight of an object, no matter where we put the axis of rotation, the magnitude of torque m*g*X is always as if the whole mass of a body is pulling from center of mass. I simply can not figure out why that is

Even if looking at formula

m*g*X=(x1*m1+x2*m2 ... )*g

I get no clues what so ever


I know that we can treat extended objects as if the whole body mass was concentrated at that point(center of mass), but ...

Something just doesn't add up


I will give an example

If we have two balls m1 and m2 (m2 > m1) connected together with a stick with length L (its weight is neglible), we find center of that object's mass with

r - is a distance of ball with mass m1 from center of mass

r*m1*g = (l - r)*m2*g

This sounds reasonable


But we could also find center of mass if we don't put rotation axis on center of mass but instead anywhere else. In that case center of mass would be a point where

X*m*g = x1*m1*g + x2*m2*g

This I don't understand! I understand that sum of M1 and M2 would cause the same effect as both those torques together, but why this also gives us a center of mass?

thank you

 

[Doc Al]

But we could also find center of mass if we don't put rotation axis on center of mass but instead anywhere else. In that case center of mass would be a point where

X*m*g = x1*m1*g + x2*m2*g

This I don't understand! I understand that sum of M1 and M2 would cause the same effect as both those torques together, but why this also gives us a center of mass?

Think of each mass exerting its own torque about the axis (which could be anywhere). The total torque is just the sum of the individual torques, which is the right hand side of that equation. Now ask: Is there a location such that if all the mass of the object where located there we would get the same torque? Find that location by setting up the above equation and solving for X. The answer is that $X = (x_1 m_1 + x_2 m_2)/m$, which is the definition of center of mass!

So you can conclude that whenever you need to find the torque produced by the weight of an extended object, you can get the answer by replacing the object by a point mass (equal to the mass of the object) located at the object's center of mass.

 

[xailer]

No, you misread my post. I already understood everything you explained! What I don't understand is how or why no matter where the axis is the magnitude of torque m*g*X is always as if the whole mass of a body is pulling from center of mass. I realize that

$X = (x_1 m_1 + x_2 m_2)/m$

always gives us the distance from axis to center of mass, but why is that? Why not a distance just a bit shorter then one to center of mass ?

 

[Doc Al]

I already understood everything you explained! What I don't understand is how or why no matter where the axis is the magnitude of torque m*g*X is always as if the whole mass of a body is pulling from center of mass.

But I thought that was exactly what I just explained! The sum of the individual torques add up to Mg times the distance to the center of mass: a mathematical truism. (Of course, we are only speaking of the horizontal coordinate of the center of mass.)

 

[xailer]

But I thought that was exactly what I just explained! The sum of the individual torques add up to Mg times the distance to the center of mass: a mathematical truism.

But WHY does formula always give you the distance to the center of a mass ? So far I only have your word (and book's )that no matter where the axis is, the torque will always be as if the whole mass is at the center of mass

 

[Doc Al]

No need to take anyone's word for it. We derived the result! Is there something about the derivation that you aren't sure of?