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Asked by arup.isro | 19 Dec, 2018, 10:52: PM
Let X1 , X2 and X3 are the x-coordinate of respective masses 1kg, 2kg and 3kg  of Centre of mass of first sytem.
Let Y1 , Y2 and Y3 are the y-coordinate of respective masses 1kg, 2kg and 3kg  of Centre of mass of first sytem.
Let Z1 , Z2 and Z3 are the z-coordinate of respective masses 1kg, 2kg and 3kg  of Centre of mass of first sytem.

we have , X1 + 2X2 + 3X3 = 6  ............................(1)
Y1 + 2Y2 + 3Y3 = 12 .............................(2)
Z1 + 2Z2 + 3Z3 = 18 ..............................(3)

Let X4 and X5 are the x-coordinate of respective masses 3kg and 2kg  of Centre of mass of second sytem.
Let Y4 and Y5 are the y-coordinate of respective masses 3kg and 2kg  of Centre of mass of second sytem.
Let Z4 and Z5 are the z-coordinate of respective masses 3kg and 2kg  of Centre of mass of second sytem.

we have,  3X4 + 2X5 = -5 ............................ (4)
3Y4 + 2Y5 = 15 .........................(5)
3Z4 + 2Z5 = -10  ......................(6)

Let X6 , Y6 and Z6 are the coordinates of 5 kg mass to be placed to get the Centre of mass same as that of first system.

If X-coordinate of Center of mass of new system is 1 then we have  : X1 + 2X2 + 3X3 + 3X4 + 2X5 +5X6 = 16
using eqns.(1) and (4),  6 -5 + 5X6 = 16  or  X6 = 3

If Y-coordinate of Center of mass of new system is 2 then we have  : Y1 + 2Y2 + 3Y3 + 3Y4 + 2Y5 +5Y6 = 32
using eqns.(2) and (5),  12 +15 + 5Y6 = 32  or  Y6 = 1

If Z-coordinate of Center of mass of new system is 3 then we have  : Z1 + 2Z2 + 3Z3 + 3Z4 + 2Z5 +5Z6 = 48
using eqns.(1) and (4),  18-10+ 5Z6 = 48  or  Z6 = 8

Coordinates of 5 kg mass = (3, 1, 8)
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