• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Several parts of a system and its CM

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.
Homework 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.
 

Delta2

Homework Helper
Insights Author
Gold Member
2,250
623
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)##
 
Last edited:

DEvens

Education Advisor
Gold Member
975
275
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.
 

Want to reply to this thread?

"Several parts of a system and its CM" You must log in or register to reply here.

Related Threads for: Several parts of a system and its CM

Replies
1
Views
6K
Replies
2
Views
8K
  • Posted
Replies
1
Views
908
Replies
1
Views
2K
Replies
5
Views
2K
Replies
19
Views
1K
Replies
0
Views
3K
Replies
5
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top