$A$ hollow cone of radius $R$ and height $H = 2R$ is placed on an inclined plane of inclination $\theta$. If $\theta$ is increased gradually,at what value of $\theta$ will the cone topple? Assume sufficient friction is present to prevent slipping.

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:

An equilateral prism of mass $m$ rests on a rough horizontal surface with coefficient of friction $\mu$. A horizontal force $F$ is applied on the prism at the top vertex $A$ as shown in the figure. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, then the minimum force required to topple the prism is $-$

Two points $A$ and $B$ on a rigid body are moving as shown in the figure. The angular velocity of the body is:

$A$ 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