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

william1941 | Student

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)