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Moment, Center of Mass, Centroids

Author: Richard Enderton

Prerequisites

Students should be able to apply the concept of integration as a summation.

Moments and Center of Mass - One and Two Dimensional Discrete Mass Systems

This video looks at the terminology and techniques involved with finding center of mass of linear and two-dimensional systems.

Center of Mass - Planar Laminas

This video develops the integral technique for calculating the center of mass of a planar lamina (and centroid of a region).

Center of Mass, Centroids - Examples

This video walks through three example of applications of the formulas.

Problem Set

1. Find the center of mass of the given point masses lying on the x-axis

$m subscript 1 equals 7 comma space m subscript 2 equals 4 comma space m subscript 3 equals 3 comma space m subscript 4 equals 8 x subscript 1 equals negative 3 comma space x subscript 2 equals negative 2 comma space x subscript 3 equals 5 comma space x subscript 4 equals 9$ (2.636)

2. Find the center of mass of the given system of point masses

$m colon space space space space space space space space space space space space 4 space space space space space space space space space space space 2 space space space space space space space space space 3.5 space space space space space space space space space 6 left parenthesis x subscript i comma y subscript i right parenthesis colon space space space space left parenthesis 2 comma 3 right parenthesis space space space space left parenthesis negative 1 comma 5 right parenthesis space space space space left parenthesis 6 comma 8 right parenthesis space space space space left parenthesis negative 3 comma 0 right parenthesis$ (0.581, 3.226)

3. Find the center of mass of the planar lamina of uniform density bounded by

$y equals square root of x comma space y equals 0 comma space x equals 4$ (12/5, 3/4)

4. Find the center of mass of the planar lamina of uniform density bounded by

$y equals negative x squared plus 4 x plus 2 comma space y equals x plus 2$ (3/2, 22/5)

5. Find the center of mass of the planar lamina of uniform density which is a polygonal region with the vertices

(0,3), (1,3), (1,1), (3,1), (3,2), (4,2), (4,0), (0,0) (1.786, 1.071)

6. Find the center of mass of the planar lamina of uniform density which has the region shown

(0, 3.167) with x-axis along the bottom of the circles and the y-axis between the circles

7. Find the centroid of the region bounded by

$y equals x cubed comma space y equals x comma space 0 less or equal than x less or equal than 1$ (8/15, 8/21)

8. Find the centroid of the polygonal region (trapezoid) with vertices

(0,0), (0,4), (5,4), (9,0) (3.595,1.810)

Additional Resources

Paul's Online Notes

http://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx

Online Physics Lab

http://dev.physicslab.org/Document.aspx?doctype=3&filename=RotaryMotion_CenterMass.xml

Math Open Source

http://www.mathopenref.com/trianglecentroid.html

Math Words

http://pballew.net/centroid.html

Wolfram Alpha