$A$ 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.

$A$ 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).

$A$ 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:

$A$ 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 ............. $kg$.

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.

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 $-$