Four particles each of mass $m$ are lying symmetrically on the rim of a disc of mass $M$ and radius $R$. The moment of inertia of the system about an axis passing through one of the particles and perpendicular to the plane of the disc is:

$A$ particle of mass $M=0.2 \ kg$ is initially at rest in the $xy$-plane at a point $(x=-l, y=-h)$,where $l=10 \ m$ and $h=1 \ m$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10 \ m/s^2$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin,in $SI$ units,are represented by $\vec{L}$ and $\vec{\tau}$,respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x, y$ and $z$-directions,respectively. If $\hat{k}=\hat{i} \times \hat{j}$,then which of the following statement$(s)$ is(are) correct?
$(A)$ The particle arrives at the point $(x=l, y=-h)$ at time $t=2 \ s$.
$(B)$ $\vec{\tau}=2 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$.
$(C)$ $\vec{L}=4 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$.
$(D)$ $\vec{\tau}=\hat{k}$ when the particle passes through the point $(x=0, y=-h)$.

State whether the following statements are true or false:
$(1)$ Torque produces angular velocity in an object.
$(2)$ For the rotational motion of a rigid body,the linear variables of all its particles are the same.

$A$,$B$,and $C$ are a disc,a solid sphere,and a spherical shell respectively,with the same radii $(R)$ and masses $(M)$. These bodies are placed as shown in the figure. The moment of inertia of the given system about the axis $PQ$ is $\frac{x}{15} I$,where $I$ is the moment of inertia of the disc about its diameter. The value of $x$ is . . . . . . .

The portion $AB$ of the wedge shown in the figure is rough and $BC$ is smooth. $A$ solid cylinder rolls without slipping from $A$ to $B$. The ratio of translational kinetic energy to rotational kinetic energy,when the cylinder reaches point $C$,is:

$A$ hinged construction consists of three rhombs with the side ratio $5:3:2$. Vertex $A_3$ moves in the horizontal direction at a velocity $v$. The velocity of $A_2$ is ....... $V$.