Understanding Center of Mass: Exploring the Concept of Vector Mass and Position

In summary, the equations provided in the forum post define and expand on the concept of center of mass, showing its dependence on total mass and individual masses in a system.
  • #1
deda
185
0
This is attempt to address Arcon's page on
center of mass

He says: [tex]\vec{R}=\frac{\sum m_i\vec{r_i}}{M}[/tex]
R is radius vector of the center;
ri are radius vectors of mi;
M is sum of all mi;

It can be generalized to vector mass too.

[tex]M\vec{R}=M\vec{e_M}\times\vec{R}\times\vec{e_M}=\vec{M}\times\vec{R}\times\vec{e_m}[/tex];

also just like
[tex]\vec{M}=\sum \vec{m_i}[/tex]
under special conditions it can as well be
[tex]\vec{R}=\sum \vec{r_i}[/tex]

=> [tex](\sum \vec{r_i})\times(\sum \vec{m_i})=\sum (\vec{r_i}\times\vec{m_i})[/tex];

=> [tex]\sum_{i<>j} (\vec{r_i}\times\vec{m_j})=\vec{o}[/tex];

This way the position of the last mass depends on all the masses (including its own) and all the other positions;

I just can't say what are the terms for the last equation.
Can you?
 
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  • #2




Thank you for bringing up this important topic of center of mass. I would like to clarify and expand on the equations provided in the forum post.

The first equation, \vec{R}=\frac{\sum m_i\vec{r_i}}{M}, is known as the definition of center of mass. It represents the average position of all the masses in a system, weighted by their individual masses. This equation can be generalized to include vector masses as well, as shown in the second equation M\vec{R}=M\vec{e_M}\times\vec{R}\times\vec{e_M}=\vec{M}\times\vec{R}\times\vec{e_m}. Here, \vec{e_M} and \vec{e_m} represent the unit vectors in the direction of the total mass and vector mass, respectively.

The third equation, \vec{M}=\sum \vec{m_i}, is called the definition of total mass. It simply states that the total mass of the system is equal to the sum of all the individual masses. Under special conditions, the fourth equation \vec{R}=\sum \vec{r_i} can also hold true, where \vec{r_i} represents the position vector of the individual masses.

The last equation, \sum_{i<>j} (\vec{r_i}\times\vec{m_j})=\vec{o}, is known as the vector cross product. It represents the sum of all the cross products between the position vectors of the individual masses and the vector masses of all the other masses in the system. This equation shows that the position of the last mass is dependent on all the masses and their positions in the system.

I hope this clarifies the terms used in the forum post. If you have any further questions, please feel free to ask. As scientists, it is important for us to have a clear understanding of the concepts we discuss. Keep exploring and asking questions!
 

1. What is the definition of center of mass?

The center of mass is a point in an object or system where the mass is evenly distributed, meaning the object will balance at that point when suspended.

2. How is center of mass different from center of gravity?

Center of mass takes into account the distribution of mass within an object, while center of gravity only considers the gravitational forces acting on an object. The center of gravity is usually close to the center of mass, but may differ depending on the strength and direction of external forces.

3. How is the center of mass calculated?

The center of mass is calculated by taking the weighted average of the positions of all the individual particles that make up an object. This takes into account both the mass and position of each particle, resulting in a single point that represents the center of mass.

4. What is the significance of center of mass in physics?

The center of mass is a crucial concept in physics as it helps determine the overall motion and stability of an object or system. It is also used in various equations and laws, such as Newton's second law of motion and the law of conservation of momentum.

5. How does the concept of vector mass contribute to understanding center of mass?

Vector mass takes into account the direction and magnitude of each particle's mass, rather than just its position. This allows for a more precise calculation of the center of mass, especially for irregularly shaped objects. It also helps in understanding the rotational motion of an object around its center of mass.

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