Centroid Planar Lamina

Centroids in a Planar Lamina
Young Lee
Brian Lau
Alvin Lau
Jonathan Javier

Introduction to Centroids
Centroid (Center of gravity)
The center of mass of an object of uniform density
Characteristics of a centroid
If a region has a line of symmetry, then its centroid is on that line of symmetry
If a region has two lines of symmetry, then the centroid is the point where they intersect
Moment
A measure of tendency to cause a body to rotate about a specific point
No movement at the center of gravity
Planar Lamina
A thin and flat plate of material of constant density
Density: measure of mass per unit of area, denoted by greek letter rho (p)

Centroidal Equations
Center of Mass (centroid):
Mass of lamina:
Moment of planar lamina about the x-axis:
Moment of planar lamina about the y-axis:

Illustration
f(x)
R
g(x)
a
b

Illustration
Mass of Planar Lamina (R = area of Planar Lamina)
m = (p)(R)

Illustration
f(x)
𝜟x
R
f(x) — g(x)
Subinterval, mi
g(x)
a
b

Illustration
Mass of subinterval

Illustration
Moment of mi about the x-axis
Moment of mi about the y-axis

Illustration
f(x)
𝜟x
R
f(x) — g(x)
g(x)
a
b

Centroidal Equations
Center of Mass (centroid):
Mass of lamina:
Moment of planar lamina about the x-axis:
Moment of planar lamina about the y-axis:

Example
Find the centroid of the region bounded by the graphs f(x) = x and g(x) = x4
Centroid: (0.56, 0.37)

Centroids: Real Life Application
Balance
Balancing thin objects on a point (2-D)
Balancing two people on a see-saw (1-D)
Infrastructure
Knowing the centroid of parts of buildings ensures buildings to be stable (3-D)
Crimes
Mapping the density of crimes in a certain area
Locomotion
Locating the centroid of a vehicle to increase stability

Centroid Planar Lamina

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