Mathematics
Mathematics
Problem
A uniform thin plate has the shape of an isosceles triangle with equal sides 30 cm long that meet at the top and a 20 cm base that lies horizontally. Which statement best describes the location of the plate’s center of mass? a) On the triangle’s line of symmetry, halfway between the base and the top vertex b) On the triangle’s line of symmetry, two-thirds of the distance from the base toward the top vertex c) On the triangle’s line of symmetry, one-third of the distance from the base toward the top vertex d) At the midpoint of the baseIntuitive explanation
The center of mass of any uniform shape lies at its geometric centroid—the "average" position of all its points. For a triangle, you don't need calculus: the centroid is always located at the intersection of the three medians, which is of the way up from the base to the opposite vertex. Since your isosceles triangle has its base horizontal at the bottom, you're finding how far up from that base the centroid sits, measured along the vertical line of symmetry.
Intuitive explanation
The center of mass of any uniform shape lies at its geometric centroid—the "average" position of all its points. For a triangle, you don't need calculus: the centroid is always located at the intersection of the three medians, which is of the way up from the base to the opposite vertex. Since your isosceles triangle has its base horizontal at the bottom, you're finding how far up from that base the centroid sits, measured along the vertical line of symmetry.