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.

$A$ ladder of length $L$ is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment,the speed of the end in contact with the horizontal floor is $v$ and the ladder makes an angle $\alpha = 30^o$ with the horizontal. Then the speed of the ladder's center must be

$A$ cubical block of side $L$ rests on a rough horizontal surface with coefficient of friction $\mu$. $A$ horizontal force $F$ is applied on the block as shown. If the coefficient of friction is sufficiently high so that the block does not slide before toppling,the minimum force required to topple the block is

Calculate the minimum amount of work necessary to overturn a crate of mass $100 \ kg$,first about edge $AB$,then about edge $A_1B_1$. The dimensions of the crate are given in the figure. (Assume $g = 10 \ m/s^2$) (in $J$)

$A$ 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:

$A$ 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