Center of Mass
- What is the center of mass?
- What are the center of mass equations for a finite number of particles?
- What do all those variables mean and why do we use subscripts?
A mass weighted average of all the locations of particles in a system. The center of mass usually coincides with the center of gravity, or the balance point of the system unless the value of the acceleration due to gravity changes significantly through the system. (i.e. very tall buildings) The center of mass is the location at which the force of gravity appears to act upon an object if all of that object’s mass were concentrated at that point.
Location of the x-coordinate:
Location of the y-coordinate:
Location of the z-coordinate:
Where the total mass of the system is
And N is the total number of particles in the system.
Each variable represents a quantity or value (i.e. mass, x-coordinate, and so forth). The subscripts are used to help with bookkeeping. If you need to refer to 3 different masses, the first one is called m1, the second one is m2, and the third one is m3. It is also helpful to use the subscripts in a consistent manner to keep track of the variables. Here is an example of a system of 4 particles:
i |
Mass symbol |
Mass Value (kg) |
X-Position symbol |
X -Position (m) |
Y-Position symbol |
Y -Position (m) |
1 |
m1 = |
2.00 |
x1 = |
20 |
y1 = |
10 |
2 |
m2 = |
5.00 |
x2 = |
-20 |
y2 = |
5 |
3 |
m5 = |
3.00 |
x3 = |
20 |
y3 = |
-10 |
4 |
m4 = |
10.00 |
x4 = |
-20 |
y4 = |
-5 |