The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $V\left(r\right)=k{r}^2/2$, where $k$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $R$ about the point $O$. If $v$ is the speed of the particle and $L$ is the magnitude of its angular momentum about $O$, which of the following statements is (are) true?

A solid sphere of radius $R$ has a moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$. If it's moment of inertia about the tangential axis (which is perpendicular to the plane of the disc), is also equal to $I$, then the value of $r$ is equal to

A solid sphere is in pure rolling motion on an inclined surface having inclination $\mathrm{\theta }$, the friction force acting on sphere will be,

A ball moves over a fixed track as shown in the figure. From $A \text{to} B$ the ball rolls without slipping. If surface $BC$ is frictionless and $K_A, K_B \text{and} K_C$ are kinetic energies of the ball at $A, B \text{and} C$, respectively, then

A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\frac{3v^2}{4g}$ with respect to the initial position. The object is 

Statement-I: Two cylinders, one hollow (metal) and the other solid (wood), with the same mass and identical dimensions, are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first.

and

Statement-2: By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

A thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At times, $t=0$, a small insect starts from $O$ and moves with a constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at $t=T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $\left(\left|\tau \right|\right)$ on the system about $O$, as a function of time, is best represented by which plot?

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion is the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by ${\overrightarrow{L}}_{\mathrm{O}}$ and ${\overrightarrow{L}}_{\mathrm{P}}$, respectively, then