Three particles of masses 50 g, 100 g and 150 g are placed at the vertices of an equilateral triangle of side 1 m (as shown in the figure). The (x, y) coordinates of the centre of mass will be:

Concept:

A point relative to the system of number of particles in an object is defined as centre of mass. The centre of mass of x is given by the formula:

\({{\rm{X}}_{{\rm{COM}}}} = \frac{{{{\rm{x}}_{\rm{a}}}{{\rm{m}}_{\rm{a}}} + {{\rm{x}}_{\rm{b}}}{{\rm{m}}_{\rm{b}}} + {{\rm{x}}_{\rm{c}}}{{\rm{m}}_{\rm{c}}}}}{{{{\rm{m}}_{\rm{a}}} + {{\rm{m}}_{\rm{b}}} + {{\rm{m}}_{\rm{c}}}}}\)

Where,

x_a, x_b, and x_c are x coordinates of the position of particles

m_a, m_b, and m_c are mass of the particles.

Calculation:

From question, we need to find the venter of mass of the three particles.

Let the three particles be a, b, and c.

First, we need to find the coordinate of a, b, and c.

⇒ (x, y)_a = (0, 0)

⇒ (x, y)_b = (1, 0)

\(\because\left[ {\begin{array}{*{20}{c}} {\tan 60 = \frac{{{{\rm{y}}_{\rm{c}}}}}{{0.5}}}\\ { \Rightarrow \sqrt 3 \times 0.5 = {{\rm{y}}_{\rm{c}}}}\\ {\therefore {{\rm{y}}_{\rm{c}}} = \frac{{\sqrt 3 }}{2}} \end{array}} \right]\)

\( \Rightarrow {\left( {{\rm{x}},{\rm{y}}} \right)_{\rm{c}}} = \left( {0.5,\frac{{\sqrt 3 }}{2}} \right) = \left( {\frac{1}{2},\frac{{\sqrt 3 }}{2}} \right)\)

The centre of mass of x is given by the formula:

\({{\rm{X}}_{{\rm{COM}}}} = \frac{{{{\rm{x}}_{\rm{a}}}{{\rm{m}}_{\rm{a}}} + {{\rm{x}}_{\rm{b}}}{{\rm{m}}_{\rm{b}}} + {{\rm{x}}_{\rm{c}}}{{\rm{m}}_{\rm{c}}}}}{{{{\rm{m}}_{\rm{a}}} + {{\rm{m}}_{\rm{b}}} + {{\rm{m}}_{\rm{c}}}}}\)

On substituting the vales,

\( \Rightarrow {{\rm{X}}_{{\rm{COM}}}} = \frac{{50\left( 0 \right) + 100\left( 1 \right) + 150\left( {\frac{1}{2}} \right)}}{{50 + 100 + 150}}\)

\( \Rightarrow {{\rm{X}}_{{\rm{COM}}}} = \frac{{100 + 75}}{{300}}\)

\( \Rightarrow {{\rm{X}}_{{\rm{COM}}}} = \frac{{175}}{{300}}\)

\(\therefore {{\rm{X}}_{{\rm{COM}}}} = \frac{7}{{12}}{\rm{\;m}}\)

The centre of mass of y is given by the formula:

\({{\rm{Y}}_{{\rm{COM}}}} = \frac{{{{\rm{y}}_{\rm{a}}}{{\rm{m}}_{\rm{a}}} + {{\rm{y}}_{\rm{b}}}{{\rm{m}}_{\rm{b}}} + {{\rm{y}}_{\rm{c}}}{{\rm{m}}_{\rm{c}}}}}{{{{\rm{m}}_{\rm{a}}} + {{\rm{m}}_{\rm{b}}} + {{\rm{m}}_{\rm{c}}}}}\)

Where,

y_a, y_b, and y_c are y coordinates of the position of particles

m_a, m_b, and m_c are mass of the particles.

On substituting the vales,

\( \Rightarrow {{\rm{Y}}_{{\rm{COM}}}} = \frac{{50\left( 0 \right) + 100\left( 0 \right) + 150\left( {\frac{{\sqrt 3 }}{2}} \right)}}{{50 + 100 + 150}}\)

\( \Rightarrow {{\rm{Y}}_{{\rm{COM}}}} = \frac{{75\left( {\sqrt 3 } \right)}}{{300}}\)