The figure above shows three uniform objects a rod with m1

The figure above shows three uniform objects: a rod with m1 = 4.50 kg, a right triangle with m2 = 2.20 kg, and a square with m3 = 6.10 kg. Their coordinates in meters are given. Determine the center of gravity for the three-object system.

Solution

for m1 :

centre of mass will be in middle of rod.

(x1, y1) = ( ( 2 + (9-7)/2 ), 7 ) = ( 2+3.5, 7 ) = (5.5, 7 ) m

for m2:

centre of mass of triangle is at 2h/3 from two end of hypotenuse

(x2, y2) = ( (4 + (2* (8-4)/3)), (1 + ((5-1)/3)) ) = ( 6.67, 2.33 )

for m3:

(x3, y3) = ( - (2 + (5-2)/2), (2 + (5-2)/2)) = ( - 3.5, 3.5 )

ccentre of mass = (m1r1 + m2r2 + m3r3) / (m1 + m2 + m3)

Xcm = [ (4.50* 5.5) + ( 2.20 * 6.67) + (6.10* -3.5) ] / (4.50 + 2.20 + 6.10 )

   = 1.41 m

Ycm = [ (4.50* 7) + ( 2.20 * 2.33) + (6.10* 3.5) ] / (4.50 + 2.20 + 6.10 )

   = 4.53 m

center of gravity = (1.41, 4.53) m