Question

Find the position of centre of mass of the uniform planner sheet shown in the figure with respect to the origin(O).

• A. $a/6,a/6$
• B. $3a/2,3a/2$
• C. $5a/6,5a/6$
• D. $a,a$

Answer 1

The correct option is C $5a/6,5a/6$

Let $\sigma$ is mass per unit area. Let Centre of mass (COM) in x direction is $x$ and COM in y direction $y$

$x=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}$

$y=\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3}{m_1+m_2+m_3}$

As shown in figure, $m_A=\sigma \times a^2=m$, $m_B=\sigma \times a^2=m$, $m_C=\sigma \times a^2=m$

$\Rightarrow x_A=\frac{a}{2}$, $x_B=\frac{a}{2}$, $x_C=\frac{3a}{2}$

$\Rightarrow y_A=\frac{3a}{2}$, $y_B=\frac{a}{2}$, $y_C=\frac{a}{2}$

$\Rightarrow x=\frac{m\times \frac{a}{2}+m\times \frac{a}{2}+m\times \frac{3a}{2}}{m+m+m}=\frac{5a}{6}$

$\Rightarrow y=\frac{m\times \frac{3a}{2}+m\times \frac{a}{2}+m\times \frac{a}{2}}{m+m+m}=\frac{5a}{6}$

Hence , Position of COM $(\frac{5a}{6},\frac{5a}{6})$