The variation of density of a cylindrical thick and long rod, is $\rho  = {\rho _0}\frac{{{x^2}}}{{{L^2}}}$ , then position of its centre of mass from $x = 0$ end is

A firecracker is thrown with velocity of $30 \,ms ^{-1}$ in a direction which makes an angle of $75^{\circ}$ with the vertical axis. At some point on its trajectory, the firecracker splits into two identical pieces in such a way that one piece falls $27 \,m$ far from the shooting point. Assuming that all trajectories are contained in the same plane, how far will the other piece fall from the shooting point? (Take, $g=10 \,ms ^{-2}$ and neglect air resistance)

Masses  ${\rm{8, 2, 4, 2 }}kg{\rm{ }}$ are placed at the corners $A, B, C, D$ respectively of a square $ABCD$ of diagonal $80\,cm$. The distance of centre of mass from $A$ will be ........ $cm$

A rod of length $L$ has non-uniform linear mass density given by $\rho(\mathrm{x})=\mathrm{a}+\mathrm{b}\left(\frac{\mathrm{x}}{\mathrm{L}}\right)^{2},$ where $a$ and $\mathrm{b}$ are constants and $0 \leq \mathrm{x} \leq \mathrm{L}$. The value of $\mathrm{x}$ for the centre of mass of the rod is at

A circular plate of diameter ' $a$ ' is kept in contact with a square plate of side $a$ as shown. The density of the material and the thickness are same everywhere. The centre of mass of composite system will be ...........

A uniform rectangular thin sheet $ABCD$ of mass $M$ has length $a$ and breadth $b$, as shown in the figure. If the shaded portion $HBGO$ is cut off, the coordinates of the centre of mass of the remaining portion will be