Right Triangle Centroid

Calculus Level 3

Compute the y y -coordinate of the center of mass of an object in the shape of an isosceles right triangle of side lengths 2 , 2 , 2 2 2,2,2\sqrt{2} in the first quadrant with the right angle at the origin. Assume the object has constant mass area density σ = 1 \sigma = 1 .

1 6 \frac16 1 2 \frac12 2 3 \frac23 4 3 \frac43

Matt DeCross by Matt DeCross , 27, USA

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2 solutions

Matt DeCross
Apr 25, 2016

The integral defining the center of mass coordinate is:

∫ 0 2 ∫ 0 2 − x y d y d x = ∫ 0 2 ( 2 − x ) 2 2 d x = 4 3 . \int_0^2 \int_0^{2-x} y dy dx = \int_0^2 \frac{(2-x)^2}{2} dx = \frac{4}{3}.

This quantity must be normalized by the total area A = 1 2 b g = 2 A = \frac12 bg = 2 , giving a final answer of

y c o m = 1 2 4 3 = 2 3 . y_{com} = \frac12 \frac43 = \frac23.

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Bhavya Jain
Jul 7, 2016

As the linear mass density is constant so centre of mass of the traingle lies on centroid . Hence y - coordinate is 2/3

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