The centre of mass of three particles of masses 1 kg, 2 kg and 3 kg is at (3, 3, 3) with reference to a fixed coordinate system. Where should a fourth particle of mass 4 kg be placed, so that the centre of mass of the system of all particles shifts to a point (1, 1, 1)?

Question

The centre of mass of three particles of masses 1 kg, 2 kg and 3 kg is at (3, 3, 3) with reference to a fixed coordinate system. Where should a fourth particle of mass 4 kg be placed, so that the centre of mass of the system of all particles shifts to a point (1, 1, 1)? (A)  (-1,-1,-1)  (B)  (-2,-2,-2)  (C)  (2, 2, 2)  (D)  (1, 1, 1)

Tardigrade Admin · Accepted Answer

Centre of mass of a solid body is given by

x_{CM} = \frac{i = 1 \sum n Δ m _{i} x _{i}}{j = 1 \sum n Δ M _{j}}, y_{CM} = \frac{i = 1 \sum n Δ m _{i} y _{i}}{j = 1 \sum n Δ M _{j}}

z_{CM} = \frac{i = 1 \sum n Δ m _{i} z _{i}}{j = 1 \sum n Δ M _{j}}

1 \times x_{1} + 2 \times x_{2} + 3 \times x_{3} = (1 + 2 + 3) 3

..(i)
and

x_{1} = x_{2} = x_{3} = 3

x_{CM} = y_{CM} = z_{CM} = 1 (g i v e n)

1 (1 + 2 + 3 + 4) = 1 x_{1} + 1 x_{2} + 3 x_{3} + 4 x_{4}

..(ii)
Solving Eqs, (i) and (ii),
we get

4 x_{4} = 10 - 18 \Rightarrow x_{4} = - 2

Similarly,

y_{4} = - 2, z_{4} = - 2

The fourth particle must be placed at the point

(- 2, - 2, - 2) .

Q. The centre of mass of three particles of masses 1 kg, 2 kg and 3 kg is at (3, 3, 3) with reference to a fixed coordinate system. Where should a fourth particle of mass 4 kg be placed, so that the centre of mass of the system of all particles shifts to a point (1, 1, 1)?

$(- 1, - 1, - 1)$

$(- 2, - 2, - 2)$

$(2, 2, 2)$

$(1, 1, 1)$

Q. The centre of mass of three particles of masses 1 kg, 2 kg and 3 kg is at (3, 3, 3) with reference to a fixed coordinate system. Where should a fourth particle of mass 4 kg be placed, so that the centre of mass of the system of all particles shifts to a point (1, 1, 1)?

(−1,−1,−1)

(−2,−2,−2)

(2,2,2)

(1,1,1)

$(- 1, - 1, - 1)$

$(- 2, - 2, - 2)$

$(2, 2, 2)$

$(1, 1, 1)$