Question

Half of the rectangular plate shown in figure is made of a material of density $p_1$ and the other half of density $p_2$. The length of the plate is $L$. Locate the centre of mass of the plate.

• A. ${\left[\frac{p_1+4p_2}{3(p_1+p_2)}\right]}^2$
• B. ${\left[\frac{2p_1+3p_2}{5(p_1+p_2)}\right]}^2$
• C. ${\left[\frac{p_1+3p_2}{4(p_1+p_2)}\right]}^2$
• D. ${\left[\frac{2p_1+5p_2}{3(p_1+2p_2)}\right]}^2$

Answer 1

The correct option is C

${\left[\frac{p_1+3p_2}{4(p_1+p_2)}\right]}^2$

The centre of mass of each half is located at the geometrical centre of that half. Thus, the left half may

be replaced by a point particle of mass $K_{p_1}$ placed at $C_1$ and the right half may be replaced by a point

particle of mass $K_{p_2}$ placed at $C_2$. This replacement is for the specific purpose of locating the combined

centre of mass. Take the middle point of the left edge to be the origin. The x-coordinate of $C_1$ is $\frac{L}{4}$ and

that of $C_2$ is $\frac{3L}{4}$. Hence, the x-coordinate of the centre of mass is
$x=\frac{(K_{p_1}{)}^{\frac{L}{4}}+(K_{p_2}{)}^{\frac{3L}{4}}}{K_{p_1}+K_{p_2}}$
$=\frac{(p_1+3p_2)}{4(p_1+p_2)}L$.
The combined centre of mass is this much to the right of the assumed origin.