$A$ 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 of mass $m$ 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 time $t = T$ 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?

  • A
    $A$ plot showing a linear increase of torque with time for $t < T$ and zero for $t > T$.
    Option A
  • B
    $A$ plot showing a constant torque for $t < T$ and zero for $t > T$.
    Option B
  • C
    $A$ plot showing a parabolic increase of torque with time for $t < T$ and zero for $t > T$.
    Option C
  • D
    $A$ plot showing a linear decrease of torque with time for $t < T$ and zero for $t > T$.
    Option D

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$A$ thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement$(s)$ is/are correct,when the rod makes an angle $60^{\circ}$ with vertical? [$g$ is the acceleration due to gravity]
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$(4)$ The normal reaction force from the floor on the rod will be $\frac{Mg}{16}$

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$A$ solid sphere of radius $R$ has a moment of inertia $I$ about its geometric axis. It is melted and recast into a disc of radius $r$ and thickness $t$. If the moment of inertia of this disc about an axis tangent to its edge and perpendicular to its plane is also $I$,find the value of $r$.

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