Section 15.10
Applications: Center of Mass
Where does the object account balance?
1
Introduction
If an object has uniform density, its Center of Mass is its geometric centroid.
If the density varies, we use multiple integrals to weight the positions.
2
Formulas
Moments
- Mass: .
- Moment about yz-plane: .
- Moment about xz-plane: .
- Moment about xy-plane: .
3
Visualizing Center of Mass
Interactive: Hemisphere Centroid
4
Worked Examples
Example 1: Centroid of a Triangle
Find the centroid of the triangle with vertices .
1. Area (Mass with \(\rho=1\)):
.
Hypotenuse line: .
2. Moments:
.
.
.
.
3. Centroid:
.
.
Example 2: Centroid of a Hemisphere
Find for .
1. Mass (Volume):
.
2. Moment Mxy:
.
Rho: .
Phi: .
Theta: .
.
3. Result:
.
Example 3: Moment of Inertia
Find for a cylinder with uniform density .
1. Setup:
.
.
2. Solve:
z-int: .
r-int: .
theta-int: .
3. Substitute Mass:
Mass .
.
5
Practice Quiz
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