Four particles \(A, B, C\) and \(D\) with masses \(m_A=m, m_B=2m, m_C=3m\) and \(m_D=4m\) are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is

\({\overrightarrow a _C} = a\hat i\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,\,\,\,\,{\overrightarrow a _D} =  - a\hat j\)

\({\overrightarrow a _{cm}} = \frac{{{m_a}{{\overrightarrow a }_a}a + {m_b}{{\overrightarrow a }_b} + {m_c}{{\overrightarrow a }_c} + {m_d}{{\overrightarrow a }_d}}}{{{m_a} + {m_b} + {m_c} + {m_d}}}\)

\({\overrightarrow a _{cm}} = \frac{{ - ma\hat i + 2ma\hat j + 3ma\hat i - 4ma\hat j}}{{10m}}\)

\( = \frac{{2ma\hat i - 2ma\hat j}}{{10m}} = \frac{a}{5}\hat i - \frac{a}{5}\hat j = \frac{a}{5}\left( {\hat i - \hat j} \right)\)