Center of Mass of Uniform Symmetric Bodies

[andyrk]

Why are centre of mass of all uniform symmetric bodies at there geometric centre?
We know the following result:
"Consider a system of point masses m₁,m₂,m₃... located at co-ordinates (x₁,y₁,z₁), (x₂,y₂,z₂)...respectively. The centre of mass of this system of masses is a point whose co-ordinates are ((x_cm,y_cm,z_cm))
Which are given by
x_cm=(m₁x₁+m₂x₂...)/(m₁+m₂+m₃...)

y_cm=(m₁y₁+m₂y₂...)/(m₁+m₂+m₃...)

z_cm=(m₁z₁+m₂z₂...)/(m₁+m₂+m₃...)

How did we get this result?

 

[voko]

This is not a result, this is a definition.

 

[dreamLord]

That comes from the definition of the center of mass.

EDIT : Apologies, already posted above.

 

[andyrk]

Why are so many things in Physics simply defined out of nowhere? Like Centre of Mass, Kinetic Energy, Relation between gravitational potential energy and work done, moment of inertia, linear and angular momentum, work etc..?

 

[voko]

They are not defined out of nowhere.

Is you real question "why is CM defined that way"?

 

[andyrk]

I had posted another question too. Why is centre of mass of all uniform and symmetric bodies at their geometric centre?

 

[andyrk]

They are not defined out of nowhere.

Is you real question "why is CM defined that way"?

Yes. And why are all other quantities defined the way they are?

 

[voko]

When you have many point masses, $m_1, \ m_2, \ ... \ m_n$, which interact with some forces from one to another, Newton's third law says that for each force there is one that is exactly the opposite of it, so the sum of all the forces is zero. Because forces are on the right hand side of Newton's second law, the sum of the left hand sides must also be zero, resulting in $m_1 \vec{a}_1 + m_2 \vec{a}_2 + \ ... \ + m_n \vec{a}_n = 0$. If, in addition to the forces of interaction, there are some external forces, then we have $m_1 \vec{a}_1 + m_2 \vec{a}_2 + \ ... \ + m_n \vec{a}_n = \vec{F}_e$, where the right hand side is the sum of the external forces.

Using the definition of the CM, we have $\vec{r}_{CM} = \frac {m_1 \vec{r}_1 + m_2 \vec{r}_2 + \ ... \ + m_n \vec{r}_n} {m_1 + m_2 + \ ... \ + m_n}$, and its acceleration is then $\vec{a}_{CM} = \frac {m_1 \vec{a}_1 + m_2 \vec{a}_2 + \ ... \ + m_n \vec{a}_n} {m_1 + m_2 + \ ... \ + m_n}$, and the previous result yields $$ m\vec{a}_{CM} = \vec{F}_e $$

where $m = m_1 + m_2 + \ ... \ + m_n$, i.e., the total mass.

So we can replace a system of interacting masses with a single point mass located at the CM of the system, and it will move exactly in the same way. When we do not really care about the motions of the individual point masses, as is the case with solid bodies, this simplifies things quite a bit. And even if we do care, the simple motion of the CM helps, too.

 

[namanjain]

that's just a bit of common sense we can apply!
consider the rod
______________________

there are a collection of particles with mass 'dm'

x(cm)= ∫xdx
____
∫dm

solving by taking mass 'dm' at x distance from rod


_________dm_______
distance x

solve the integration or refer anything from webpage on ---- center of mass of continuous bodies.

 

[pongo38]

If the centre of mass was not at the geometric centre of a uniform body, it would fall over in a gravitational field. We would all be lying on the floor. (Some of us are, actually, and our potential would be fairly limited)