Two masses are placed in x-y plane. 1 kg mass is at (6, 6) and 2 kg is placed at (x, y). Find the value of (x, y) if the centre of mass of two masses is at origin.

CONCEPT:

• Center of mass: Centre of the mass of a body is a point at which the whole of the mass of the body appeared to be concentrated.
• The position of the center of mass of a three-body system in the x-direction is calculated by:

\(⇒ x_{com}= {m_1 x_1+m_2 x_2 \over m_1+m_2}\)

\(⇒ y_{com}= {m_1 y_1+m_2 y_2 \over m_1+m_2}\)

where xcom is the position of the center of mass in x-coordinate, ycom is the position of the center of mass in y-coordinate, m1, m2, and m3 are the masses of three bodies, x1, x2, and x3 are the position of different masses in x-coordinate, y1, y2, and y3 are the position of different masses in y-coordinate.

CALCULATION:

Given that (x1,y1) = (6, 6); (x2,y2) = (x, y) cm; m1 = 1 kg, m2 = 2 kg;  (xcom,ycom) = (0, 0) 

• Position of the center of mass in x - axis:

\(⇒ x_{com}= {m_1 x_1+m_2 x_2 \over m_1+m_2}\)

\(⇒ 0={1 \times 6 + 2 \times x \over 1+2}\)

⇒ x = - 3

Position of the center of mass in x - axis

\(⇒ y_{com}= {m_1 y_1+m_2 y_2 \over m_1+m_2}\)

\(⇒ 0={1 \times 6 + 2 \times y \over 1+2}\)

⇒ y = - 3

• So the position of 2^nd mass will be (x, y) = (-3, -3). Hence the correct answer is option 3.