$A$ hollow sphere of radius $R$ and mass $m$ is fully filled with water of mass $m$. It is rolled down a horizontal plane such that its centre of mass moves with a velocity $v$. If it purely rolls

  • A

    Kinetic energy of the sphere is $\frac{5}{6}\, mv^2$

  • B

    Kinetic energy of the sphere is $\frac{4}{5}\, mv^2$

  • C

    Angular momentum of the sphere about a fixed point on ground is $\frac{8}{3}\, mvR$

  • D

    Angular momentum of the sphere about a fixed point on ground is $\frac{14}{5}\, mvR$

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A thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time, $t =0$, a small insect starts from $O$ and moves with 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 $(|\vec{\tau}|)$ on the system about $O$, as a function of time is best represented by which plot?

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