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 $x=2m,y=0$ The centre of the square plate is at $x=y=0$. Find the coordinates of the centre of mass of the remaining piece

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

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
$0=\frac{\left(\sigma \cdot 2^2\right)(2)+\sigma \cdot \left(6^2-2^2\right)\left(x_R\right)}{\sigma \cdot 6^2}=\frac{8+32x_R}{36}$
$\Rightarrow x_R=\frac{-8}{32}=-\frac{1}{4}m$
Similarly, $y_B=\frac{m_L\cdot y_L+m_R\cdot y_R}{m_L+m_R}$
$0=\frac{\left(m_2\right)0+m_R\left(y_R\right)}{m_L+m_R}$
$\Rightarrow y_R=0$