Several parts of a system and its CM

In summary, the conversation discusses a theorem used for solving problems involving calculating the center of mass of a given shape. The theorem was initially thought to be obvious and was not known to be provable. However, it can be proved by using the additive property of volume integrals. This property states that if a volume can be broken down into non-overlapping sub-volumes, then the integral of a vector function over the entire volume is equal to the sum of integrals over each sub-volume. The conversation also mentions a specific case involving two masses and then introducing a third mass, and it is shown that the order in which the masses are considered does not affect the outcome. The general proof of the theorem is then discussed.
  • #1
brotherbobby
696
163
Homework Statement
Prove that the center of mass (CM) of a system composed of several parts can be determined by assuming that all the parts are particles located at their (respective) center of mass.
Relevant Equations
The CM of a system of particles each having masses ##m_i## and position vectors ##\mathbf{r_i}## is given by : ##\mathbf{r_{\text{CM}}} = \frac{\Sigma m_i \mathbf{r_i}}{\Sigma m_i}##
I have known and used this theorem for a long time solving problems ("Calculate the CM of the some given shape"). I took the theorem to be "obvious" and didn't know it could be proved (and that indeed it was a theorem at all).

I can make no attempt at the proof. Any help would be welcome.
 
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  • #2
You can prove it easily by using the additive property of volume integrals. More specifically if we have a volume ##V##, that we can break to sub volumes ##V_1,V_2,...,V_n## such that the volumes ##V_i## do not overlap and such that ##V=V_1+V_2+...V_n## then the following holds:
$$\int_V \vec{f}(\vec{r})d^3\vec{r}=\int_{V_1}\vec{f}(\vec{r})d^3\vec{r}+\int_{V_2}\vec{f}(\vec{r})d^3\vec{r}+...+\int_{V_n}\vec{f}(\vec{r})d^3\vec{r}$$
The volume ##V## is the volume of the big system and the volumes ##V_i## are the volumes of the system's parts.
If ##V>V_1+V_2+...+V_n## then we also need that ##\vec{f}(\vec{r})=0## for ##\vec{r}\in V-(V_1+V_2+...+V_n)##
 
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  • #3
Try it explicitly for the simple case of two masses in one group and one mass in the other group. So you have ##D_1 = \frac{m_1R_1 + m_2R_2}{m_1+m_2}## and then you bring in ##m_3## located at ##R_3##. Show that it does not matter which order you do it, either all three as one big system, or first the first two then the third. After that it's a question of how you proceed to a general proof.
 

Related to Several parts of a system and its CM

1. What is a CM or Configuration Management of a system?

Configuration Management (CM) is the process of managing and controlling changes made to a system over its lifetime. It involves identifying, tracking, and managing all components and their relationships within a system, ensuring that changes are properly evaluated, approved, and implemented.

2. What are the main components of a system that need to be managed through CM?

The main components of a system that need to be managed through CM include hardware, software, documentation, and data. These components are interdependent and require careful management to ensure that the system functions properly and efficiently.

3. Why is CM important for a system?

CM is important for a system because it helps to ensure that the system remains stable and functional throughout its lifetime. It also allows for easier and more efficient maintenance and updates, reduces the risk of errors and failures, and improves overall system quality and performance.

4. How does CM help to prevent errors or conflicts in a system?

CM helps to prevent errors or conflicts in a system by maintaining a record of all changes made to the system and ensuring that they are properly evaluated and approved before implementation. This helps to prevent unauthorized or conflicting changes that could lead to errors or system failures.

5. What are some common tools used for CM?

Some common tools used for CM include version control systems, such as Git or SVN, which track changes to source code and allow for collaboration among developers. Other tools include configuration management databases (CMDBs) and automated deployment systems, which help to manage and deploy changes to a system in a controlled and organized manner.

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