A uniform square plate and a disc having same mass per unit area are kept in contact as shown in Fig. The side of square and diameter of circle are both equal to `L`. Locate the position of centre of mass of the system w.r.t. the centre of the square.

Explanation

Let mass per unit area of the square plate = mass per unit area of disc `= sigma`
`:.` Mass of square plate
`m_(1) = sigmaL^(2)`, and mass of disc
`m_(2) = (pir^(2)) sigma = (pi(L^(2))/(4)) sigma`
Centre of square is `O_(1)`, where `m_(1)` is concentrated and centre of disc is `O_(2)`, where `m_(2)` is concentrated. If `x` is distance of c.m of the combination from `O_(1)`, then
`x_(cm) = (m_(1)x_(1) + m_(2)x_(2))/(m_(1) + m_(2)) = (m_(1) xx 0 + m_(2)L)/(m_(1) + m_(2))`
`=((pi(L^(2))/(4))sigma xx L)/(sigmaL^(2) + (pi L^(2)sigma)/(4)) = (sigmapi L^(2)//4 xx L)/(sigmaL^(2)(1 + pi//4))`
`x_(cm) = (pi//4 xx L)/((4 + pi)//4) = ((piL)/(4 + pi))`