1. Mass Center
EF Week 13

2. Generalized Moment
The first moment of “anything” about a point O is the product of two things:
the position (vector) from O to the “anything” and
the “anything” itself.

3. FMM for a Particle
The first moment of a particle’s mass, m, about O is the product of m and its position relative to O. This is called the particle’s first mass moment.
If a particle of mass 5 kg is located at position cm, what is the particle’s first mass moment about O?

4. Properties of FMM The FMM is not a vector product (dot, cross).
The FMM is a vector.
The dimensions of FMM are mass times length.

5. FMM: Multiple Particles
For a collection of multiple particles, the first moment about O is the sum of the individual particles’ first moments about O.

6. Mass Center
At the mass center of a system of particles, the first mass moment of the system is zero.
If O is the mass center, what does that tell us?

7. Where is the mass center of a system with more than two particles?
The fact that the FFM is zero at the mass center is not enough information to find the mass center.
To find the mass center, we need the following definition:
The mass center is the point where we could put the total mass of the system without changing the system’s FMM.

8. Let G denote the mass center. Then from the definition of mass center,
This gives us a formula for finding the position of the mass center.
The position is an average value (a weighted sum divided by an unweighted sum).

9. Locate the mass center of the three-particle system shown (coordinates in cm).

10. Mass Center Properties
The position of the mass center (PMC) is the average position of all the mass.
The PMC is the FMM of the system divided by the total mass.
At the mass center, the FMM is zero.
If planes of symmetry exist, the mass center is in them.

11. Rigid Body
A rigid body can be thought of as a collection of an infinite number of particles, each with infinitesimal mass, rigidly “glued” together.
Then, to find the first mass moment of a rigid body, we must sum an infinite number of moments!
But you have encountered infinite sums before in calculus.

12. FMM: Rigid Body
In the limit as the number of particles becomes infinite and the largest particle becomes infinitesimal, the first mass moment becomes an integral.
The m below the integral denotes the domain over which the integral is evaluated, which is the entire mass of the body.

13. Mass Center: Rigid Body

14. Centroid: Rigid Body
If a rigid body has a uniform mass density, then its mass center and centroid coincide.

15. Locate the mass center of the uniformly dense rectangular parallelepiped shown.

16. Composite Bodies - 1
Because the integral of a sum equals the sum of the integrals, we can separate integrals in into pieces, evaluate the pieces, and then add the pieces back together.
Sometimes we can evaluate the pieces from simple formulas in tables.

17. Composite Bodies - 2 Identify the simple shapes in the body.
Write the FMM of the entire body as the sum of the FMM’s of the simple shapes.
Replace the FMM of each simple shape with the product of its mass and the position of its mass center.
Add the FMM of the simple shapes and divide by the total mass.

18. Locate the mass center for a uniformly dense mallet modeled as the stick-cube composite body shown.
From symmetry,

20. Where is the mass center of the uniformly thick angle iron?

22. Where is the mass center?