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:

$A$ block with a square base measuring $a \times a$ and height $h$ is placed on an inclined plane. The coefficient of friction is $\mu$. The angle of inclination $(\theta)$ of the plane is gradually increased. The block will:

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.

$A$ 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:

$A$ rod of length $l$ is held vertically stationary with its lower end located at a point $P$ on a horizontal plane. When the rod is released to topple about $P$, the velocity of the upper end of the rod with which it hits the ground is