Question

A uniform square plate of side 6 m has had a square piece of side 2 m cut out of it as shown in figure. The centre of that piece is at $x=2\, m , y=0 $ The centre of the square plate is at $x= y=0$. Find the coordinates of the centre of mass of the remaining piece

• A. $-2\, m ,-2 \,m$
• B. $-1 \,m ,-1 \,m$
• C. $-1 \,m , 0 \,m$
• D. $\frac{-1}{4} m , 0 \,m$

Answer 1

Answer: (D)

$B$ -Big square, $L$ -Little square,
$R$ - Remaining piece
The big square is made up of the little square and the remaining piece
Hence for centre of mass coordinates,
$x_{B}=\frac{m_{L} \cdot x_{L}+m_{R} \cdot x_{R}}{m_{L}+m_{R}}$
Let $\sigma$ be mass per unit area, then
$0=\frac{\left(\sigma \cdot 2^{2}\right)(2)+\sigma \cdot\left(6^{2}-2^{2}\right)\left(x_{R}\right)}{\sigma \cdot 6^{2}}=\frac{8+32 x_{R}}{36}$
$\Rightarrow x_{R}=\frac{-8}{32}=-\frac{1}{4} \,m$
Similarly, $y_{B}=\frac{m_{L} \cdot y_{L}+m_{R} \cdot y_{R}}{m_{L}+m_{R}}$
$0=\frac{\left(m_{2}\right) 0+m_{R}\left(y_{R}\right)}{m_{L}+m_{R}} $
$\Rightarrow y_{R}=0$