The instantaneous velocity of point $B$ of a rod of length $0.5 \ m$ is $3 \ m/s$ at an angle of $30^{\circ}$ with the rod as shown. Find the angular velocity of the rod such that the velocity of end $A$ is minimum.

$A$ cube with sides of length $2a$ and mass $M$ is moving with an initial speed $v_0$ along a frictionless table. When the cube reaches the end of the table it is caught abruptly by a short lip and begins to rotate. The minimum speed $v_0$ such that the cube falls off the table is

$A$ disc is rolling (without slipping) on a horizontal surface. $C$ is its center and $Q$ and $P$ are two points on the disc such that $Q$ is further from the contact point $O$ than $C$,and $P$ is closer to the contact point $O$ than $C$. Let $V_P, V_Q$ and $V_C$ be the magnitudes of the velocities of points $P, Q$ and $C$ respectively,then:

$A$ cubic block of side $L$ rests on a rough surface with a sufficient coefficient of friction. $A$ horizontal force $F$ is applied at the top edge of the block. If the coefficient of friction is high enough so that the block does not slide before it topples,the minimum force required to topple the block is:

$A$ disc rolls on a horizontal surface (without slipping). $C$ is the center and $Q$ and $P$ are two points at the same distance from $C$. Let $v_P$,$v_Q$,and $v_C$ be the magnitudes of the velocities of points $P$,$Q$,and $C$ respectively. Then:

$A$ table has a heavy circular top of radius $1 \,m$ and mass $20 \,kg$ placed on four light (considered massless) legs placed symmetrically on its circumference. The maximum mass that can be kept anywhere on the table without toppling it is close to ............. $kg$.