Question

The centre of mass of three bodies each of mass 1 kg located at the points (0, 0), (3, 0) and (0, 4) in the XY plane is
A. \[\left( {\dfrac{4}{3},1} \right)\]
B. \[\left( {\dfrac{1}{3},\dfrac{2}{3}} \right)\]
C. \[\left( {\dfrac{1}{2},\dfrac{1}{2}} \right)\]
D. \[\left( {1,\dfrac{4}{3}} \right)\]

Answer 1

Hint:Before we start addressing the problem, we need to know what data has been provided. They have given the coordinates in the XY plane and three bodies having 1kg mass each. In order to solve this, we need the formula for the centre of mass.

Formula Used:
The formula to find the centre of mass is given by,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3}}}{{{m_1} + {m_2} + {m_3}}}\]
Where, \[{m_1},{m_2},{m_3}\] are the masses of three particles and \[{x_1},{x_2},{x_3}\] are the positions of masses.

Complete step by step solution:
Now, consider the centre of mass of three bodies in which each having a mass of 1 kg located at the points (0, 0), (3, 0) and (0, 4) in the XY plane then the centre of mass \[{X_{CM}}\] is given by,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3}}}{{{m_1} + {m_2} + {m_3}}} \\ \]
According to the question, we have \[{m_1} = {m_2} = {m_3} = 1\,kg\], \[\left( {{x_1},{y_1}} \right) = \left( {0,0} \right)\] , \[\left( {{x_2},{y_2}} \right) = \left( {3,0} \right)\] and \[\left( {{x_3},{y_3}} \right) = \left( {0,4} \right)\]
\[{X_{CM}} = \dfrac{{1\left( 0 \right) + 1\left( 3 \right) + 1\left( 0 \right)}}{{1 + 1 + 1}} \\ \]
\[\Rightarrow{X_{CM}} = \dfrac{3}{3} \\ \]
\[ \Rightarrow {X_{CM}} = 1 \\ \]
Similarly,
\[\Rightarrow{Y_{CM}} = \dfrac{{{m_1}{y_1} + {m_2}{y_2} + {m_3}{y_3}}}{{{m_1} + {m_2} + {m_3}}} \\ \]
\[\Rightarrow{Y_{CM}} = \dfrac{{1\left( 0 \right) + 1\left( 0 \right) + 1\left( 4 \right)}}{{1 + 1 + 1}} \\ \]
\[ \therefore {Y_{CM}} = \dfrac{4}{3}\]
Therefore, the coordinates of the centre of mass are \[\left( {1,\dfrac{4}{3}} \right)\].

Hence, option D is the correct answer.

Note: The point in the system which will move in the same way that a single particle would move when it is subjected to an external force is called the centre of mass of the system. Or the centre of mass is the point at which the whole mass of the body is concentrated. The centre of mass of a body can be located in the middle of the object or at some distance from the centre. For example, a bottle will have its centre of mass on its axis.