Centre of Mass of Isosceles Triangle?

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SUMMARY

The discussion focuses on determining the center of mass (COM) of an isosceles triangle with base 'a' and height 'h' using integration techniques. The triangle is defined as a two-dimensional continuous structure with uniform density and negligible thickness. The approach involves slicing the triangle into horizontal strips parallel to the X-axis, where the mass of each strip is calculated as the product of density and area. The variable of integration is 'y', which represents the height, and the length 'l' of each strip must be expressed as a function of 'y'.

PREREQUISITES
  • Understanding of two-dimensional continuous structures
  • Knowledge of integration techniques in calculus
  • Familiarity with concepts of center of mass
  • Basic geometry of triangles, specifically isosceles triangles
NEXT STEPS
  • Study integration methods for calculating center of mass in various shapes
  • Learn how to derive functions of length in relation to height for geometric figures
  • Explore applications of density in mass calculations for continuous bodies
  • Investigate the properties of isosceles triangles in more complex scenarios
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Students in physics or engineering courses, educators teaching mechanics, and anyone interested in applications of calculus to geometric problems.

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Homework Statement


an isosceles with base=a, height=h. It has uniform density and negligible thickness.


Homework Equations


determine it's centre of mass by integration,using more than one orientation
PHP:

The Attempt at a Solution



It is a two-dimensional continuous structure. Let the base,a, lie on the X-axis and the Y-axis
be axis of symmetry. Then the COM must lie on the Y-axis. Slice the triangle into strips parallel to the X-axis. Each strip is of width dy, and length l. The mass per strip,dm, is obtained through:

mass=density x area,

p l dy
and the variable of integration is y as it is the axis of symmetry and integration is over 0 and h.
 
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You will need to express l in terms y, since l is a function of y.
 

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