Question

A system consists of three particles, each of mass m and located at (1,1) (2,2) and (3,3). Then find the coordinates of the center of the mass.
A. (1,1)
B. (2,2)
C. (3,3)
D. (6,6)

Answer 1

Hint:Before we start addressing the problem, we need to know about the system of particles. A system of particles is a group of particles that are interrelated. The equations for a system of particles can be used to develop those for a rigid body. One new but very important concept introduced with a system of particles is the concept of the centre of mass. 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.

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.
\[{x_1},{x_2},{x_3}\] are the positions of masses.

Complete step by step solution:
Consider a system consisting of three particles, each of mass m which is located at (1,1) (2,2) and (3,3). Then we need to find the coordinates of the centre of the mass.
In order to find the coordinates, we have,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3}}}{{{m_1} + {m_2} + {m_3}}}\]
Here, they have given, \[{m_1} = {m_2} = {m_3} = m\] then,
\[{X_{CM}} = \dfrac{{m\left( {{x_1} + {x_2} + {x_3}} \right)}}{{3m}}\]
\[\Rightarrow {X_{CM}} = \dfrac{{\left( {{x_1} + {x_2} + {x_3}} \right)}}{3}\]

Similarly, \[{Y_{CM}} = \dfrac{{\left( {{y_1} + {y_2} + {y_3}} \right)}}{3}\]
Using the coordinates given as (1,1) (2,2) and (3,3) we have,
\[{X_{CM}} = \dfrac{{\left( {1 + 2 + 3} \right)}}{3}\]
\[ \Rightarrow {X_{CM}} = 2\]
Then,
\[{Y_{CM}} = \dfrac{{\left( {1 + 2 + 3} \right)}}{3}\]
\[ \therefore {Y_{CM}} = 2\]
Therefore, the coordinates of the centre of the mass are (2,2).

Hence, Option B is the correct answer

Note:Centre of mass of the system of particles helps us in describing the behaviour of a macroscopic body in terms of the laws developed for microscopic particles.