ELECTROSTATICS: An Equilateral Triangle’s Center of Mass ( Part 2 )

In a prior example, a visually engaging technique was used to locate the center of mass within an equilateral triangle:

A more mathematically detailed approach will now be used to determine the center of mass location. The diagram above must be expanded in such a manner that trigonometry can be applied to our efforts:

The rays that connect the diagonal sides of the triangle to their respective vertices ( corners ) are of particular interest. These rays are called medians, and they split the 60⁰ angles at each vertex into 30⁰ angles. 

The a̅e̅ ray is the height ( h ) of a right triangle situated within the larger equilateral triangle, and it is opposite to the 60⁰ angle of the leftmost vertex. We use this relationship to rewrite the height in terms of the length of the triangle’s sides:

sin θ = ( a̅e̅ / d )

h = a̅e̅ = d sin θ

sin 60⁰ = √3/2

h = a̅e̅ = d√3/2

Recall that a̅o̅ ray is ⅔ h ( a̅e̅ ), so ray o̅e̅ is ⅓ h:

o̅e̅ = ⅓ h = ⅓ a̅e̅ = ( ⅓ )( d sin 60⁰ ) = ( ⅓ )( d√3/2 ) = d√3/6

The value of the hypotenuse of the obe triangle will give us the center of mass ( com ) distance relative to the lowermost vertices of the triangle:

sin θ = ( o̅e̅ / com )

sin 30⁰ = ( o̅e̅ / com )

sin 30⁰ = ½ 

com = ( o̅e̅ / sin 30⁰ ) = [ ( d√3/6 ) / ( ½ ) ] = d√3/3

For an equilateral triangle of side length ( d ), the center of mass is a distance of d√3/3 meters relative to the lowermost vertices. Relative to the uppermost vertex, the center of mass location is as follows:

( ⅔ )( h ) = ( ⅔ )( a̅e̅ ) = ( ⅔ )( d√3/2 ) = d√3/3