$A$ ring of mass $m$ and radius $R$ has three particles attached to the ring as shown in the figure. The centre of the ring has a speed $v_0$. The kinetic energy of the system is: (Slipping is absent) (in $, mv_0^2$)

In the case of rotational dynamics,which one of the following statements is correct?
$[\vec{\omega} = \text{angular velocity}, \vec{v} = \text{linear velocity}, \vec{r} = \text{radius vector}, \vec{\alpha} = \text{angular acceleration}, \vec{a} = \text{linear acceleration}, \vec{L} = \text{angular momentum}, \vec{p} = \text{linear momentum}, \vec{\tau} = \text{torque}, \vec{f} = \text{force}]$

Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side length $d$. The ratio $I_{0} / I_{A}$ of the moment of inertia $I_{0}$ of the system about an axis passing through the centroid and perpendicular to the plane of the triangle,to the moment of inertia $I_{A}$ about an axis passing through the center of any one of the spheres and perpendicular to the plane of the triangle,is:

$A$ bar of mass $M=1.00 \ kg$ and length $L=0.20 \ m$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. $A$ small mass $m=0.10 \ kg$ is moving on the same horizontal surface with $5.00 \ m \ s^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed $v$. After this elastic collision,the bar rotates with an angular velocity $\omega$. Which of the following statements is correct?

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 $m$ is moving along the side of a square of side '$a$',with a uniform speed $v$ in the $x-y$ plane as shown in the figure. Which of the following statements is false for the angular momentum $\vec L$ about the origin?