Calculus I · Unit 3B · lesson
Moments and Center of Mass
Learn Moments and Center of Mass through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Length, surface, mass, and balanceWhat this section is building
Learn Moments and Center of Mass through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Arc length adds short chord lengths; surface area adds narrow bands; mass adds density-times-size; moments add position-weighted mass.
Identify the local size element first, then multiply by circumference, density, or position only when the modeled quantity requires it.
Using geometric midpoint for a nonuniform object, forgetting the surface radius, or treating differential legs as independent finite lengths.
Learning objectives
Compute the first moment and center of mass of a one-dimensional object.
Moments and Center of Mass
Center of mass is a weighted average of position
A center of mass is not generally the geometric midpoint. Pieces with more mass exert more influence on the balance point. The moment about an origin multiplies each small mass contribution by its position, and integrating produces the total weighted position. Dividing the moment by total mass gives the center of mass.
The choice of origin affects the moment but not the physical balance point after the coordinates are interpreted consistently. Symmetry can simplify calculations and should be used before integration. A sensible final answer must lie within the object's occupied interval or region. If it falls outside, either the moment sign, density, or total mass has been mishandled.
For density , mass is
and the moment about the origin is
The center of mass is
Center of mass of the increasing-density rod
For on ,
Thus
This lies to the right of the midpoint , as expected because density increases with .
u3b-com-01For density on , find the center of mass.
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Compute mass and moment, then divide.
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Balance point of a nonuniform rod. A nonuniform rod balances at x-bar, shifted toward the denser side. Show density, weighted moment, and center of mass location. Do not imply center of mass must be geometric midpoint.
Every relationship in balance point of a nonuniform rod uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show density, weighted moment, and center of mass location.
Balance point of a nonuniform rod. Show density, weighted moment, and center of mass location.
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