The identical spheres each of mass $2 \mathrm{M}$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $4 \mathrm{~m}$ each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $\frac{4 \sqrt{2}}{x}$, where the value of $x$ is_____

For a body, centre of volume is defined as $\frac{{\int {r.dV} }}{{\int {dV} }}$ over complete body, where $dV$ is small volume of body and $\vec r$ is. position vector of that small volume from origin

An object comprises of $a$ uniform ring of radius $R$ and its uniform chord $AB$ (not necessarily made of the same material) as shown. Which of the following can not be the centre of mass of the object

$A$ small ball $B$ of mass $m$ is suspended with light inelastic string of length $L$ from $a$ block $A$ of same mass $m$ which can move on smooth horizontal surface as shown in the figure. The ball is displaced by angle $\theta$ from equilibrium position & then released. The displacement of centre of mass of $A+ B$ system till the string becomes vertical is

The position vector of the centre of mass $\vec r\, cm$ of an asymmetric uniform bar of negligible area of cross-section as shown in figure is

The coordinates of the positions of particles of mass $7,\,4{\rm{  and 10}}\,gm$ are ${\rm{(1,}}\,{\rm{5,}}\, - {\rm{3),}}\,\,{\rm{(2,}}\,5,7{\rm{) }}$ and ${\rm{(3, 3, }} - {\rm{1)}}\,cm$ respectively. The position of the centre of mass of the system would be