Three identical spheres each of mass $M$ are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to $3\,m$ each. Taking the point of intersection of mutually perpendicular sides as the origin,the magnitude of the position vector of the centre of mass of the system will be $\sqrt{x}\,m$. The value of $x$ is

Two bodies $A$ and $B$ have masses $M$ and $m$ respectively,where $M > m$,and they are at a distance $d$ apart. Equal forces are applied to each of them so that they approach each other. The position where they hit each other is

The centre of mass of an extended body on the surface of the earth and its centre of gravity:

Define the centre of mass.

Look at the drawing given in the figure which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles,and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6m$. The coordinates of the centres of the different parts are: outer circle $(0,0)$,left inner circle $(-a, a)$,right inner circle $(a, a)$,vertical line $(0,0)$ and horizontal line $(0,-a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is

Two smooth and similar right-angled prisms are arranged on a smooth horizontal plane as shown in the figure. The lower prism has a mass $3$ times the upper prism. The prisms are held in an initial position as shown and are then released. As the upper prism touches the horizontal plane,the distance moved by the lower prism is