A uniform rod hinged at its one end is allowed to rotate in vertical plane. Rod is given an angular velocity $\mathrm{\omega }$ in its vertical position as shown in the figure. The value of $\mathrm{\omega }$ for which the force exerted by the hinge on the rod is zero in this position is:

   

Two points $A$ and $B$ on a disc have velocities $v_1$ and $v_2$ at some moment. Their directions make angles $60°$ and $30°$, respectively, with the line of separation as shown in the figure. The angular velocity of the disc is

Uniform rod $AB$ is hinged at the end $A$ in a horizontal position as shown in the figure (the hinge is frictionless, that is, it does not exert any friction force on the rod). The other end of the rod is connected to a block through a massless string as shown. The pulley is smooth and massless. Masses of the block and the rod are same and are equal to $m.$ Acceleration due to gravity is $g.$ The tension in the thread and angular acceleration of the rod just after the release of block from this position.

In the figure shown a ring $A$ is rolling without sliding with a velocity $v$ on the horizontal surface of the body $B$ (of the same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height (from initial position) reached by $A$ on $B$.

A uniform rod of mass $m$, length $l$ is placed over a smooth horizontal surface along $y$-axis and is at rest as shown in the figure. An impulsive force $F$ is applied for a small-time $∆t$ along $x$-direction at the point $A$ after this rod moves freely. The $x$–coordinate of the end $A$ of the rod when the rod becomes parallel to $x$–axis for the first time is (initially the coordinate of centre of mass of the rod is $\left(0, 0\right)$

A uniform rectangular plate of mass $m$ which is free to rotate about the smooth vertical hinge passing through the centre and perpendicular to the plate is lying on a smooth horizontal surface. A particle of mass $m$ moving with speed $u$ collides with the plate and sticks to it as shown in the figure. The angular velocity of the plate after the collision will be

A particle of mass $m$ is moving horizontally at speed $v$ perpendicular to a uniform rod of length $d$ and mass $M=6m$. The rod is hinged at centre $O$ and can freely rotate in the horizontal plane about a fixed vertical axis passing through its centre $O$. The hinge is frictionless. The particle strikes and sticks to the end of the rod. The angular speed of the system just after the collision

A uniform sphere of mass $m$ is given some angular velocity about a horizontal axis through its centre and gently placed on a plank of mass $m$. The coefficient of friction between the two is $\mu$. The plank rests on a smooth horizontal surface. The initial acceleration of the centre of the sphere relative to the plank will be