System of Parti...

The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its midpoint and perpendicular to its length is $$0_0$$. Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is :

The ratio of the radii of gyration of a circular
disc to that of a circular ring, each of same mass
and the radius around their respective axes is -

Two small balls $$A$$ and $$B$$ are made of same material, $$A$$ is solid but $$B$$ is hollow. These are connected by an ideal spring and this system is kept over a smooth plank such that center of mass of two balls coincide with mid point of the plank. Initially the spring is compressed by a certain amount and the plank is in horizontal position. Now we release the system then, 

Find the momentum of each particle.

A chain $$AB$$ of mass $$m$$ and length $$L$$ is hanging on a smooth horizontal table as shown in the figure. If it is released from the position shown then the displacement of centre of mass of chain in magnitude, when end $$A$$  moves a distance $$\dfrac{L}{2}\ is\ X\sqrt{2}m $$.
Find $$X$$. ($$L = 32\  m$$)

Two bars of masses $$m_1$$ and $$m_2$$, connected by a weightless spring of stiffness $$k$$, rest on a smooth horizontal plane. Bar 2 is shifted by a small distance $$x_0$$ to the left and released. The velocity of the centre of mass of the system when bar 1 breaks off the wall is :

Two identical bricks of length $$L$$ are piled one on top of the other on a table as shown in the figure. The maximum distance $$S$$ the top brick can overhang the table with the system still balanced is :

Three point like equal masses $$m_1, m_2$$ and $$m_3$$ are connected to the ends of a mass-less rod of length $$L$$ which lies at rest on a smooth horizontal plane. At $$t = 0$$, an explosion occurs between $$m_2$$ and $$m_3$$, and as a result, mass $$m_3$$ is detached from the rod, and moves with a known velocity $$v$$ at an angle of $$30^o$$ with the y-axis. Assume that the masses $$m_2$$ and $$m_3$$ are unchanged during the explosion.
What is the velocity of the centre of mass of the system consisting of three masses after the expulsion?

The arrangement shown in figure above consists of two identical uniform solid cylinders, each of mass $$m$$, on which two light threads are wound symmetrically. The friction in the axle of the upper cylinder is assumed to be absent. If the tension of each thread in the process of motion is $$\displaystyle T=\frac{mg}{x}$$, then the value of $$x$$ is :

A particle of mass $$'m'$$ is rigidly attached at $$'A'$$ to a ring of mass $$'3m'$$ and radius $$'r.'$$ The system is released from rest and rolls without sliding. The angular acceleration of ring just after release is