1. The system consist of three objects.
Three balls are located on “x” axis as shown in the picture, find the center of gravity for the system.
Break it down:
Three balls
are located on “x” axis
as shown in the picture, find the center of gravity for the system.
The system consist of three objects.
All of objects lie on the “x” axis and system is in equilibrium state. When a system is in equilibrium state it either does not move or move with constant speed.
For a rigid system center of gravity is a point that we can assume all forces (here weight) applied to the system are applied to this point
A rigid system is a system that masses do not move relative to each other. In other words the distances between all masses in the system is constant.
Physicsfix.com

2. The total force applied to this system is [(m1 + m2 + m3) * g]
Three balls are located on “x” axis as shown in the picture, find the center of gravity for the system.
Solution:
Draw a diagram
m2 = 1kg
m1 = 4kg
m3 = 3kg
1.0m
2.0m
m2g
m1g
m3g
The total force applied to this system is [(m1 + m2 + m3) * g]
Physicsfix.com

3. Choose a coordinate system (x-y)
Three balls are located on “x” axis as shown in the picture, find the center of gravity for the system.
+ Y
Solution:
We have to find a point that we can assume this force is applied to that. This point is center of gravity.
Choose a coordinate system (x-y)
When the system is in equilibrium it does not matter where we locate the axes. To simplify, it’s better to locate the origin (0, 0) on one of the masses. Let’s put the origin on m2.
- X
+ X -Y
Physicsfix.com

4. Three balls are located on “x” axis as shown in the picture, find the center of gravity for the system.
Solution:
Determine known parameters
m1 = 4kg X1 = -1m negative sign shows that m1 is located on the left hand of the origin
m2 = 1kg X2 = m2 is located on the origin
m3 = 3kg X3 = +2m m3 is located on +x axis, 2m far from the origin
Calculate location of center of gravity
We use notation Xcg to show the location of center of gravity on X axis (x coordinate of center of gravity)
We calculate it by equation: Xcg = Σ (mi * Xi)/ Σ mi
Where (Σ (mi * Xi) is the sum of the produt of each mass and its distance from the origin.
Σ (mi * Xi) = m1X1 + m2X2 + m3X3 and Σ mi = m1 + m2 + m3
Physicsfix.com

5. Calculate the enumerator and denominator
Three balls are located on “x” axis as shown in the picture, find the center of gravity for the system.
Solution:
Calculate the enumerator and denominator
Σ (mi * Xi) = m1X1 + m2X2 + m3X3  Σ (mi * Xi) = [4*(-1) + 1*(0) + 3*(+2)]
Σ (mi * Xi) =
Σ mi = m1 + m2 + m3  Σ mi =
Calculate Xcg by dividing Σ (mi * Xi) to Σ mi
Xcg = Σ (mi * Xi)/ Σ mi  Xcg= +2kg.m/8kg = + 1/4m or
The positive sign shows that Xcg lies on +x axis, 0.25 m far from the origin.
 Σ (mi * Xi) = +2kg.m
 Σ mi = 8kg
Xcg = m
Physicsfix.com

6. Three balls are located on “x” axis as shown in the picture, find the center of gravity for the system.
Solution:
we can have:
These two systems are equal and it is much easier to use the second system.
Physicsfix.com