Question

Two blocks $A$ and $B$ each of equal masses $m$ are related from the top of a smooth fixed wedge as shown in the figure. Fine the magnitude of the acceleration of the centre of mass of the two blocks.

Answer 1

Letus consider incline surface of wedge as x-axis and y-axis respectively. We can write acceleration of blocks $A$ and $B$.
${\overline{a}}_A=mgsin{30}^o\hat{i}$ and ${\overline{a}}_B=mgsin{30}^o\hat{j}$
The acceleration of centre of mass of two blocks can be written as
${\overline{a}}_{cm}=\frac{m_A{\overline{a}}_A+m_B{\overline{a}}_B}{m_A+m_B}=\frac{mgsin{30}^o\hat{i}+mgsin{60}^o\hat{j}}{m+m}$
$\frac{gsin{30}^o\hat{i}+gsin{60}^o\hat{j}}{2}$
$\frac{g}{4}\hat{i}+\frac{\sqrt{3}}{4}\hat{j}⟹1{\overline{a}}_{cm}1=\frac{g}{2}$