Four particles of masses $m_1 = 2m, m_2 = 4m, m_3 = m$ and $m_4$ are placed at four corners of a square. What should be the value of $m_4$ so that the centres of mass of all the four particles are exactly at the centre of the square ?

A spherical hollow is made in a lead sphere of radius $R$, such that its surface touches the outside surface of lead sphere and passes through the centre. What is the shift in the centre of mass of lead sphere due to the hollowing?

A circular disk of moment of inertia $I_t$ is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed $\omega _i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed $\omega _f$. The energy lost by the initially rotating disc to friction is

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat i\,\, + \,\,2\hat j\,\, + \,\,\hat k$ and  $-\,3\hat i\,\, - \,\,2\hat j\,\, + \,\,\hat k$, respectively. The centre of mass of this system has a position vector

Three thin metal rods, each of mass $M$ and length $L$ are welded to form an equilateral triangle. The moment of inertia of the composite structure about an axis passing through the centre of mass of the structure and perpendicular to its plane is

A particle of mass $= 5\ units$ is moving with a uniform speed $V = 3 \sqrt 2\ units$ in the $XOY$ Plane along the line $Y = X+4 $ . The magnitude of the angular momentum of the particle about the origin is ...... $unit.$