Caption: The calculation of center of mass illustrated.

Features:

1. The center of mass is a mass-weighted average position for a physical system.

2. The system could be anything---fluid body, gas sample, set of point masses, planetary system, star cluster, galaxy, galaxy cluster---anything.

3. You divide the system up mentally into small bits (formally differential elements) and sum position vectors for the bits times the bit masses and divide by the sum of the bit masses (which is the mass of the system).

   The result is the center of mass---more explicitly the center-of-mass position vector.

   Recall, a vector is a quantity with both a magnitude and a direction.

4. The formula for center of mass is

     r⃗_cm = ∑_i (m_i*r⃗_i) / ∑_i (m_i) = ∑_i (m_i*r⃗_i) / m  

   where the supercript → signifies vector, r⃗_cm is the center of mass position vector i is an index for the bits, r⃗_i is the position vector for bit i, m_i is the mass of bit i, and m is the system mass (i.e., the total mass).

5. The importance of the center of mass of a system is actually the position variable that enters into Newton's 2nd law of motion (AKA F=ma):

     F⃗_net_ext = m*a⃗_cm  , 

   where F⃗_net_ext is the net external force on the system, m is again the system mass, and a⃗_cm is the center-of-mass acceleration.

   So it's center-of-mass motion which is determined by F=ma---a point often neglected in high-school physics courses.

6. Centers of mass are very useful in determining inertial frames for the analysis of both the INTERNAL and center of mass motions of a system.

   Center-of-mass (CM) inertial frames are discussed in Mechanics file: frame_basics.html and Mechanics file: frame_hierarchy_astro.html.

7. An important factoid is that center of mass of a system equals the center of mass evaluated from the center of masses of any set of subsystems of the physical system.

   Proof: Given a set of point masses i each with mass m_i and m=∑_i m_i and position vector r⃗_i, the center of mass r⃗_cm is determined by

     mr⃗_cm = ∑_i m_i*r⃗_i = ∑_j (∑_i m_i*r⃗_i)_subset_j = ∑_j m_jr⃗_cm_j ,

   where there are a general set of subsets j of point masses and m_j is the sum of the mass of the subset j. Clearly, r⃗_cm_j is the center of mass of the subset j. Since the subsets are general, the proof is complete: QED.