The centre of mass of a triangular plate

Abstract

We present a derivation for the coordinates of the centre of mass—or centre of gravity—of a homogeneous triangular plate by using scaling and symmetry. We scale the triangular plate by a factor of 2 and divide its area into four plates identical to the original. By symmetry, we assert that the centre of mass of two identical masses lies at the midpoint of the line joining their centres of mass. By relating the centres of mass of the original to those of the scaled plates, we find the coordinates of the centre of mass as the solution of an algebraic equation.

Moment of inertia through scaling and the parallel axis theorem

“…Note that the center of mass of this longer bar is a point located in the extremes of a couple of bars with length L. Then, other way to write the moment of inertia of the longer bar is by considering two bars with length L and the moment of inertia around an axis in their extremes. According to the parallel axis theorem [4], the moment of inertia for a bar of length L around an axis going through an extreme is…”

“…A few years ago we proposed a method to find out the center of mass of a right triangular plate by using scaling and the parallel axis theorem instead of explicit integrals [4][5]. Now we propose a similar method in order to get the moment of inertia of a right triangular plate and apply the result to discuss the case of a scalene triangular plate.…”

Using infinite series to calculate the centre of mass of triangular plates

“…[5] Slüsarenko et al undertook an interesting study dealing with the derivation for the centre of mass coordinates of a homogeneous triangular plate by using scaling and symmetry. [6] In this paper, we present a new approach to determine the centre of mass position of triangular plates by using scaling, symmetry and geometric infinite series. This paper might help students to understand that when we move from discrete point masses to continuous regions, we move from finite sums and arithmetic to limits of Riemann sums (integrals) and calculus.…”