How to calculate the centre of mass

[Sarah0001]

Homework Statement

"If all the masses were instead placed at the centre of mass then they must give the same resultant turning moment about any point in the system as all of the individual moments added together."

Relevant Equations

The equation shown in the image insert

1) I'm not sure I quite understand this statement, is there an example that can be given to show this statement mathematically?

2) Is there any derivation for this equation to calculate position centre of mass below?

 

[Delta2]

1) This statement basically means that the torque due to the weight of a rigid body (or a system of particles)(about any point O), is equal to the torque of the weight of an imaginary point particle that is placed in the center of mass of the rigid body (or in the center of mass of the system of particles) and the mass of this imaginary particle is equal to the mass of the rigid body (or equal to the total mass of the system of particles).

This statement can be stated and proven mathematically.

2) The equation shown at image is the definition of the center of mass of a system of point particles with masses $m_i$ located at distances $r_i$ from the origin. It cannot be derived from something else as it is the definition...

 

[Sarah0001]

Thank you!