Physics - Center of Mass Question with Solution | TestHub

The coordinates of the three masses are: $m_1=m$ at $(0,0)$, $m_2=2m$ at $(a,0)$, and $m_3=3m$ at $(a/2,\sqrt{3}a/2)$. The x-coordinate of the center of mass is $X_{CM}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}$.$$X_{CM}=\frac{m(0)+2m(a)+3m(a/2)}{m+2m+3m}=\frac{0+2ma+3ma/2}{6m}=\frac{7ma/2}{6m}=\frac{7a}{12}$$ The y-coordinate of the center of mass is $Y_{CM}=\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3}{m_1+m_2+m_3}$.$$Y_{CM}=\frac{m(0)+2m(0)+3m(\sqrt{3}a/2)}{m+2m+3m}=\frac{0+0+3\sqrt{3}ma/2}{6m}=\frac{3\sqrt{3}a}{12}=\frac{\sqrt{3}a}{4}$$ Thus, the center of mass is at $\left(\frac{7a}{12},\frac{\sqrt{3}a}{4}\right)$.