Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a = 15$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at $( 0,0 )$ , and that the sides are along the positive axes.
A) $( 6,6 )$
B) $( 6,15 )$
C) $( 5,6 )$
D) $( 15,15 )$
E) None of these

Question

Quizplus · Accepted Answer

The center of mass is at the intersection of the medians, which are drawn from the vertices to the midpoints of the opposite sides. For an isosceles right triangle, the medians are congruent and perpendicular, so the center of mass is at the midpoint of the hypotenuse. Since the hypotenuse has length $a\sqrt{2}$, the center of mass is at $(a\sqrt{2}/2, a\sqrt{2}/2)$. Substituting $a = 15$, we get $(6,6)$.

Find the Center of Mass of a Lamina in the Shape