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

Three masses are placed on the $x-$axis: $300 \, g$ at the origin,$500 \, g$ at $x = 40 \, cm$ and $400 \, g$ at $x = 70 \, cm$. The distance of the centre of mass from the origin is ....... $cm$.

Obtain the general expression for the centre of mass of a system of $n$ distributed particles in three dimensions.

Give the location of the centre of mass of a $(i)$ sphere,$(ii)$ cylinder,$(iii)$ ring,and $(iv)$ cube,each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?

$A$ point mass $m_A$ is connected to a point mass $m_B$ by a massless rod of length $l$ as shown in the figure. It is observed that the ratio of the moment of inertia of the system about the two axes $AA$ and $BB$,which are parallel to each other and perpendicular to the rod,is $\frac{I_{BB}}{I_{AA}} = 3$. The distance of the centre of mass of the system from the mass $m_A$ is

Write the difference between the centre of gravity and the centre of mass of a body.