Question

Two equilateral triangular plates are joined together to form the following figure. Density of plate $A$ is $3$ times that of plate $B$. If $COM$ lies inside the plate $A$ at a distance $h=\sqrt{3}\text{m}$ from their common side, find out the length of side of each triangular plate.

• A. $6\sqrt{3}\text{m}$
• B. $12\text{m}$
• C. $\frac{6}{\sqrt{3}}$
• D. $3\sqrt{3}\text{m}$

Answer 1

The correct option is B $12\text{m}$

Density of plate $A$ is three times of plate $B$ (Given).


Therefore, if mass of plate $B=m$
Mass of plate $A=3m$

Position of $COM$ of triangular plate will be $\frac{H}{3}$ from the base, where $H$ is the height of triangular plate.
Now consider the common side of the plates to be situated on $X$ axis.
Then, y coordinate at which $COM$ of $A$ lies is $h_1=\frac{H}{3}$
y coordinate at which $COM$ of $B$ lies is $h_2=\frac{-H}{3}$

y coordinate of $COM$ of system $h=\frac{m_1{h}_1+m_2{h}_2}{m_1+m_2}$
$=\frac{3m\times \frac{H}{3}+m\left(\frac{-H}{3}\right)}{m+3m}$
$=\frac{2H}{12}=\frac{H}{6}$
$\Rightarrow H=6h=6\sqrt{3}$ (given $h=\sqrt{3}$)
and we know, height of equilateral triangle $=\frac{\sqrt{3}}{2}\times$ side of triangle
Then, $6\sqrt{3}=\frac{\sqrt{3}}{2}\times$ side of triangle
$\Rightarrow$ Side of triangle is $12\text{m}$.