Find center of mass of planar quadrilateral

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SUMMARY

The center of mass of a planar quadrilateral with vertices (0, 0), (2, 0), (1, 1), and (0, 1) can be calculated using calculus rather than assuming equal masses at the vertices. To find the center of mass, one effective method is to divide the quadrilateral into simpler shapes, such as triangles and rectangles, and compute their individual centers of mass. The overall center of mass can then be determined by combining these values, utilizing the standard calculus formulas for area and centroid calculations.

PREREQUISITES
  • Understanding of planar geometry and quadrilaterals
  • Knowledge of calculus, specifically integration for area calculations
  • Familiarity with the concept of center of mass and centroid
  • Ability to decompose complex shapes into simpler geometric figures
NEXT STEPS
  • Study the process of calculating the center of mass for composite shapes
  • Learn about the centroid formulas for triangles and rectangles
  • Explore integration techniques for finding areas under curves
  • Review examples of calculating center of mass in physics applications
USEFUL FOR

Students in physics or engineering courses, educators teaching geometry and calculus, and anyone interested in understanding the principles of center of mass in planar shapes.

toforfiltum
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Homework Statement


Consider the planar quadrilateral with vertices (0, 0), (2, 0), (1, 1) and (0, 1). Suppose that it has constant density. What is its center of mass?

Homework Equations

The Attempt at a Solution


Since it has constant density, could I assume that the center of mass would be the same as if I put 4 equal masses on the vertices and calculate the center of mass that way? Like ##\bar x= \frac {m_1x_1+...+m_nx_n}{m_1+...m_n}## and same for ##\bar y##?

Thanks!
 
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toforfiltum said:

Homework Statement


Consider the planar quadrilateral with vertices (0, 0), (2, 0), (1, 1) and (0, 1). Suppose that it has constant density. What is its center of mass?

Homework Equations

The Attempt at a Solution


Since it has constant density, could I assume that the center of mass would be the same as if I put 4 equal masses on the vertices and calculate the center of mass that way? Like ##\bar x= \frac {m_1x_1+...+m_nx_n}{m_1+...m_n}## and same for ##\bar y##?

Thanks!
No. But you could break up the region into two pieces and put the mass of each piece at its center of mass and calculate from there.
 
LCKurtz said:
No. But you could break up the region into two pieces and put the mass of each piece at its center of mass and calculate from there.
How do I find the center of mass of the half piece?
 
toforfiltum said:
How do I find the center of mass of the half piece?
The center of mass of a rectangle is at its center and the center of mass of a triangle is at the intersection of its medians. Or you could just use the standard calculus formulas and calculate the center of mass of the region directly.
 

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