There is a rod of length $l$. The velocities of its two ends are $v_1$ and $v_2$ in opposite directions normal to the rod. The distance of the instantaneous axis of rotation from $v_1$ is:

$A$ thin rod $AB$ is sliding between two fixed right-angled surfaces. At some instant,its angular velocity is $\omega$. If $I_X$ represents the moment of inertia of the rod about an axis perpendicular to the plane and passing through the point $X$ ($A, B, C,$ or $D$),the kinetic energy of the rod is:

$A$ 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:

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.

$A$ uniform cube of side '$a$' and mass '$m$' rests on a rough horizontal surface. $A$ horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of the face,at a height $\frac{3a}{4}$ above the base. The minimum value of $F$ for which the cube begins to topple about the edge is (Assume that the cube does not slide).

$A$ 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