$A$ uniform rod $AB$ of length $L$ and mass $M$ is lying on a smooth table. $A$ small particle of mass $m$ strike the rod with a velocity $v_0$ at point $C$ a distance $x$ from the centre $O$. The particle comes to rest after collision. The value of $x$, so that point $A$ of the rod remains stationary just after collision, is :
$L/3$
$L/6$
$L/4$
$L/12$
A rod $P$ of length $1\ m$ is hinged at one end $A$ and there is a ring attached to the other end by a light inextensible thread . Another long rod $Q$ is hinged at $B$ and it passes through the ring. The rod $P$ is rotated about an axis which is perpendicular to plane in which both the rods are present and the variation between the angles $\theta $ and $\phi $ are plotted as shown. The distance between the hinges $A$ and $B$ is ........ $m.$
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 long horizontal rod has a bead which can slide along its length, and initially placed at a distance $L$ from one end $A$ of the rod. The rod is set in angular motion about $A$ with constant angular acceleration $\alpha$. If the coefficient of friction between the rod and the bead is $\mu$, and gravity is neglected, then the time after which the bead starts slipping is
The given figure shows a disc of mass $M$ and radius $R$ lying in the $x-y$ plane with its centre on $x$ axis at a distance a from the origin. then the moment of inertia of the disc about the $x-$ axis is
A disc is performing pure rolling on a smooth stationary surface with constant angular velocity as shown in figure. At any instant, for the lower most point of the disc -