Where is the center of mass of three bodies of equal mass placed at the points (0, 0), (4, 5) and (6, 3).

Where is the center of mass of three bodies of equal mass
placed at the points (0, 0), (4, 5) and (6, 3).

The center
of mass for three bodies of equal mass lies at the centroid of a triangle, at the
vertices of which lie the bodies. Here we are given the points (0, 0), (4, 5) and (6,
3).

Now to find the centroid, we need two medians and their
point of intersection. The midpoint between  (0,0) , ( 4, 5) is ( 2 , 5/2)  and the
midpoint between (0,0) and (6,3) is ( 3, 3/2)

The equation
of the line joining (2, 5/2) and (6, 3) is y-5/2 = [(3 -5/2)/
(6-2)]*(x-2)

=> y-5/2 = [(1/2)/
(4)]*(x-2)

=> 2y – 5 =
(1/4)*(x-2)

=> 2y -5 = x/4
-1/2

=> 8y –x-18 =0…
(1)

And the equation of the line joining (3, 3/2) and (4,
5) is y-3/2 = [(5 -3/2)/ (4-3)]*(x-3)

=> 2y -3 =
7*(x-3)

=> 2y – 7x + 18 =0…
(2)

(1) – 4*(2)

=> 8y –
8y – x + 28x -18 -72 =0

=> 27x =
90

=> x = 90/27=
10/3

substituting in (1)

8y =
18 + 10/3

=> y = (18 +
10/3)/8

=> y =
8

Therefore the center of mass is at (10/3,
8/3)