Proof related to the center of mass

In summary, the parallel axis theorem defines the center of mass as the position where the integral of a certain equation is equal to zero. There is no specific proof for this definition, but it can be calculated by fixing the origin at the center of mass of the body. The equation used in the theorem may vary, but the overall approach is the same.
  • #1
patric44
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Homework Statement
proof that the distribution of mass around the center of mass is = 0.
Relevant Equations
∫Rdm = 0
hi guys
in the proof of the parallel axis theorem this equation is just put as it is as a definition of the center of mass :
$$\int[2(\vec{r_{o}}.\vec{r'})I-(\vec{r_{o}}⊗\vec{r'}+\vec{r'}⊗\vec{r_{0}})]dm = 0$$
is there is any proof for this definition ? and what is the approach for it
 
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  • #2
If you calculate the position of center of mass with respect to a reference frame whose origin is fixed at the center of mass of the body then of course you get zero.

I cannot comment on your equation unless I know what are the nomenclature for the symbols but parallel axis theorem uses the setup I explained above.
 

FAQ: Proof related to the center of mass

What is the center of mass?

The center of mass is a point in an object or system where the mass is evenly distributed in all directions. It is also known as the center of gravity.

How is the center of mass calculated?

The center of mass can be calculated by finding the weighted average of the positions of all the particles in the object or system. This is done by multiplying the position of each particle by its mass and dividing the sum of these values by the total mass of the object or system.

Why is the center of mass important?

The center of mass is important because it helps us understand the stability and motion of objects. It is also a crucial concept in physics and engineering, as it is used to calculate forces and torques acting on an object.

How does the center of mass affect an object's motion?

The center of mass plays a key role in an object's motion. If an object is balanced around its center of mass, it will remain in a state of equilibrium. If the center of mass is not within the base of support, the object will topple over.

Can the center of mass be outside of an object?

Yes, the center of mass can be outside of an object. This is often the case for irregularly shaped objects or objects with holes. In these cases, the center of mass may be located in empty space or in a different material than the rest of the object.

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