Half of the recrtangular plate shown in figure is made of a material of density `rho_1` and the other half of density `rho_2`. The length of the plate is L. Locate the centre of mass of the plate.

Explanation

Assuming the distribution of mass as uniform, the centre of mass of each half plate is located at the geomertical centre of the half. Thus mass `K rho_(1)` of left half plate is at `C_(1)` and mass `K rho_(2)` of right half plate is at `C_(2)`.
Let us take middle point `O` of left edge of the plate as the origin.
Therefore, the x-coordinate of `C_(1)` is `L//4` and that of `C_(2)` is `3L//4`. Therefore, for the pull plate
`x_(cm) = (m_(1)x_(1) +m_(2)x_(2))/(m_(1) + m_(2))`
`= ((K rho_(1)) L//4 + (K rho_(2))3L//4)/(K rho_(1) + K rho_(2))`
`x_(cm) = (rho_(1) + 3rho_(2))/(4(rho_(1) + rho_(2))`
Thus centre of mass of the entire rectangular plate is at distance `x_(cm)` to the right of origin `O`.