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An isosceles triangle ABC has base width 80 mm and height 60 mm. Determine the M.I. of the triangle about its bbase

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Without loss of generality, place △ABC△ABC on the Cartesian plane with the base ABAB along the XX axis with the origin at the midpoint of of ABAB and the vertex CC on the positive YY axis.

Then, the base of the triangle, b=ABb=AB and the height of the triangle, h=OC.h=OC.

Take a small slice of the triangle of width dy,dy, parallel to AB,AB, at a distance yy above the base. Let the the length of this slice be x.x.

Since the slice is parallel to the base, it follows from the Basic Proportionality Theorem that the triangle above the slice is similar to △ABC.△ABC.

⇒xb=h−yh⇒x=bh(h−y).⇒xb=h−yh⇒x=bh(h−y).

Let the mass per unit area be σ.σ.

⇒⇒ The mass of the slice is dm=σxdy=σbh(h−y)dy.dm=σxdy=σbh(h−y)dy.

⇒⇒ The moment of inertia of the slice about the base is

dI=y2dm=σbh(h−y)y2dy