Question

Three rectangular plate of dimension each $(l\times h)$ are arranged as shown in the figure.

If the masses of plates ($A,B,C$) are in the ratio $1:2:1$, then find the coordinates of the $\text{COM}$ of the system.

• A. $(\frac{5l}{4},\frac{7h}{4})$
• B. $(\frac{5l}{4},\frac{5h}{4})$
• C. $(\frac{3l}{4},\frac{5h}{4})$
• D. $(\frac{l}{4},\frac{7h}{4})$

Answer 1

The correct option is C $(\frac{3l}{4},\frac{5h}{4})$

Coordinates of $\text{COM}$ of individual plates (considering origin at O) are:

$A(\frac{l}{2},\frac{h}{2}),B(\frac{l}{2},\frac{3h}{2}),C(\frac{3l}{2},\frac{3h}{2})$

Let the mass of plates $A,B$ and $C$ be $m,2m$ and $m$

$\Rightarrow$ Considering all the plates as point masses placed at their respective $\text{COM}$ and applying the formula for system's $\text{COM}$:

$x_{\text{CM}}=\frac{x_A{m}_A+x_B{m}_B+x_C{m}_C}{m_A+m_B+m_C}$

$x_{\text{CM}}=\frac{(m\times \frac{l}{2})+(2m\times \frac{l}{2})+(m\times \frac{3l}{2})}{m+2m+m}$

$\therefore x_{\text{CM}}=\frac{\frac{6ml}{2}}{4m}=\frac{3l}{4}$

$\Rightarrow y_{\text{CM}}=\frac{y_A{m}_A+y_B{m}_B+y_C{m}_C}{m_A+m_B+m_C}$

$y_{\text{CM}}=\frac{\frac{h}{2}.m+\frac{3h}{2}.2m+\frac{3h}{2}.m}{4m}=\frac{10hm}{8m}$

$y_{\text{CM}}=\frac{10}{8}h=\frac{5h}{4}$

$\therefore$Coordinates of $\text{COM}$ are $(\frac{3l}{4},\frac{5h}{4})$