The document discusses the differences between the center of mass and geometric center of objects. The center of mass takes into account the mass distribution of an object, while the geometric center only considers the object's geometry. The center of mass is used in physics, while the geometric center is used in design. The document also provides examples of calculating the center of mass and geometric center of different objects and systems.

1. Center of Mass
vs
Geometric Center
A presentation by Group 2

2. Differentiate Center of
mass of a system and
Geometric Center
01
OBJECTIVE

3. Center of Mass
- Is the point where the whole mass of
the body/system is concentrated
- The point where the gravitational
force, weight of the body, acts
generally in any orientation of the
body

4. Center of Mass
It is calculated using the following formula:
Center of mass
= (m1r1 + m2r2 + ... + mn rn) / (m1 +
m2 + ... + mn)

5. Geometric Center or Centroid
The geometric center is the
point where all the sides or
edges of an object intersect or
converge.

6. Geometric Center or Centroid
It is calculated using the following formula:
Geometric center
= (x1 + x2 + ... + xn) / n, (y1 + y2 + ... +
yn) / n

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8. Center of Mass vs Geometric Center
The center of mass and geometric center may
not always coincide. The geometric center is a
purely geometric property, whereas the center
of mass depends on the mass distribution of the
object. The center of mass takes into account
the mass of each object and its position, while
the geometric center is only based on the
position of each point.

9. Center of Mass vs Geometric Center
The center of mass is the point in a system
where the mass is evenly distributed, and it
behaves as if all the mass were concentrated at
that point. It takes into account the mass of
each object and its position, and it is used in
physics and engineering to analyze the motion
of systems, such as objects in freefall,
projectiles, and collisions.

10. Center of Mass vs Geometric Center
On the other hand, the geometric center is
the point where all the sides or edges of an
object intersect or converge. It is a purely
geometric property, and it is used in design,
architecture, and graphics to locate the
position of an object, such as a shape, a
building, or an image.

11. Center of Mass vs Geometric Center
While the center of mass depends on the
mass distribution of the object, the
geometric center is only based on the
position of each point or edge.
Therefore, the center of mass and geometric
center may not always coincide.

12. Problem Samples

13. Find the center of mass of a system of two
masses: 2 kg located at (1,1) and 3 kg located
at (5,3).
Center of Mass
Difficulty Level: Easy

14. Solution:
Center of mass = (2(1,1) + 3(5,3)) / (2+3) = (4,2.2)
Center of Mass
Difficulty Level: Easy

15. Explanation:
In this case, we have a system of two masses, one of
2 kg at point (1,1) and the other of 3 kg at point (5,3).
To find the center of mass, we need to use the
formula:
Center of mass = (m1r1 + m2r2) / (m1 + m2)
where m is the mass of each object and r is the
position vector of each object.
Center of Mass
Difficulty Level: Easy

16. Explanation:
Substituting the given values, we get:
Center of mass = (2(1,1) + 3(5,3)) / (2+3)
= (2, 2) + (15, 9) / 5
= (17, 11) / 5
= (3.4, 2.2)
Therefore, the center of mass of the system of two
masses is located at (3.4, 2.2).
Center of Mass
Difficulty Level: Easy

17. Find the geometric center of a rectangle with
vertices at (0,0), (0,4), (6,4), and (6,0).
Geometric Center
Difficulty Level: Easy

18. Solution:
Geometric center
= (0+0+6+6)/4, (0+4+4+0)/4 = (3,2)
Geometric Center
Difficulty Level: Easy

19. Explanation:
In this case, the opposite corners of the rectangle are
(0,0) and (6,4), and (0,4) and (6,0).
To find the geometric center, we take the average of
the x-coordinates of the corners and the average of
the y-coordinates of the corners separately.
Geometric Center
Difficulty Level: Easy

20. Explanation:
The average of the x-coordinates is
(0+0+6+6)/4 = 3, and the average of the y-
coordinates is (0+4+4+0)/4 = 2.
Therefore, the geometric center of the
rectangle is located at (3, 2).
Geometric Center
Difficulty Level: Easy

21. Find the center of mass of a system of
four masses: 2 kg located at (0,0), 4 kg
located at (4,0), 6 kg located at (4,3), and
8 kg located at (0,3).
Center of Mass
Difficulty Level: Medium

22. Explanation:
In this case, we have a system of four masses, each
with a different mass and position.
To find the center of mass, we need to use the
formula:
Center of mass
= (m1r1 + m2r2 + m3r3 + m4r4) / (m1 + m2 + m3 + m4)
where m is the mass of each object and r is the
position vector of each object.
Center of Mass
Difficulty Level: Medium

23. Explanation:
Substituting the given values, we get:
Center of mass
= (2(0,0) + 4(4,0) + 6(4,3) + 8(0,3)) / (2+4+6+8)
= (0, 0) + (16, 0) + (24, 18) + (0, 24) / 20
= (40, 42) / 20
= (3.2, 1.8)
Therefore, the center of mass of the system of four masses is
located at (3.2, 1.8).
Center of Mass
Difficulty Level: Medium

24. Find the geometric center of a regular octagon with side length
10.
The midpoint of AB is ((A + B)/2) = ((0, 5) + (5, 5))/2 = (2.5, 5).
The midpoint of BC is ((B + C)/2) = ((5, 5) + (5, 0))/2 = (5, 2.5).
The midpoint of CD is ((C + D)/2) = ((5, 0) + (0, 0))/2 = (2.5, 0).
The midpoint of DE is ((D + E)/2) = ((0, 0) + (0, 5))/2 = (0, 2.5).
The midpoint of EF is ((E + F)/2) = ((0, 5) + (-5, 5))/2 = (-2.5, 5).
The midpoint of FG is ((F + G)/2) = ((-5, 5) + (-5, 0))/2 = (-5, 2.5).
The midpoint of GH is ((G + H)/2) = ((-5, 0) + (0, 0))/2 = (-2.5, 0).
The midpoint of HA is ((H + A)/2) = ((0, 0) + (0, 5))/2 = (0, 2.5).
Geometric Center
Difficulty Level: Medium

25. Solution:
Geometric center
= [(2.5, 5) + (5, 2.5) + (2.5, 0) + (0, 2.5) + (-2.5, 5) + (-5,
2.5) + (-2.5, 0) + (0, 2.5)]/8
= (0, 2.5)
Geometric Center
Difficulty Level: Medium

26. Explanation:
To find the geometric center of this regular octagon,
we can start by finding the midpoints of each of its
sides. We can use the formula for finding the
midpoint of a line segment, which is ((x1 + x2)/2, (y1 +
y2)/2), where (x1, y1) and (x2, y2) are the coordinates
of the endpoints of the segment.
Geometric Center
Difficulty Level: Medium

27. Explanation:
Once we have found the midpoints of all eight sides
of the octagon, we can take the average of these
midpoints to find the geometric center. This average
will give us a point that is equidistant from all the
midpoints and therefore lies at the intersection of all
the lines of symmetry of the octagon.
Geometric Center
Difficulty Level: Medium

28. Explanation:
Using this method, we can calculate that the
geometric center of this regular octagon with side
length 10 is located at the
point (0, 2.5).
Geometric Center
Difficulty Level: Medium