For the pivoted slender rod of length $l$ as shown in the figure,the angular velocity as the bar reaches the vertical position after being released from the horizontal position is:

$A$ uniform thin wooden plank $AB$ of length $L$ and mass $M$ is kept on a table with its $B$ end slightly outside the edge of the table. When an impulse $J$ is given to the end $B$,the plank moves up with the centre of mass rising a distance $h$ from the surface of the table. Then,

In the case of rotational dynamics,which one of the following statements is correct?
$[\vec{\omega} = \text{angular velocity}, \vec{v} = \text{linear velocity}, \vec{r} = \text{radius vector}, \vec{\alpha} = \text{angular acceleration}, \vec{a} = \text{linear acceleration}, \vec{L} = \text{angular momentum}, \vec{p} = \text{linear momentum}, \vec{\tau} = \text{torque}, \vec{f} = \text{force}]$

Consider regular polygons with number of sides $n=3, 4, 5, \ldots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

$A$ solid sphere $(A)$ of mass $5m$ and a spherical shell $(B)$ of mass $m$, both having the same radius $R$, are placed on a rough surface. When a force $F$ of the same magnitude is applied tangentially at the highest points of $A$ and $B$, they start rolling without slipping with accelerations $a_A$ and $a_B$, respectively. The ratio of $a_A$ to $a_B$ is . . . . . . .

Three particles each of mass $m$ are located at the vertices of an equilateral triangle $ABC$. They start moving with equal speeds $v$ each along the medians of the triangle and collide at its centroid $G$. If after collision,$A$ comes to rest and $B$ retraces its path along $GB$,then $C$