According to the parallel axis theorem,$I = I_{cm} + Mx^2$. What will be the graph between $I$ and $x$?

$A$ rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power developed by the external torque as a function of time is:

$A$ solid sphere of mass $M$ and radius $R$ has a moment of inertia $I$ about its diameter. It is recast into a disc of thickness $t$ whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains $I$. The radius of the disc will be:

$A$ cord of negligible mass is wound round the rim of a flywheel of mass $20 \; kg$ and radius $20 \; cm$. $A$ steady pull of $25 \; N$ is applied on the cord as shown in the figure. The flywheel is mounted on a horizontal axle with frictionless bearings.
$(a)$ Compute the angular acceleration of the wheel.
$(b)$ Find the work done by the pull,when $2 \; m$ of the cord is unwound.
$(c)$ Find also the kinetic energy of the wheel at this point. Assume that the wheel starts from rest.
$(d)$ Compare answers to parts $(b)$ and $(c)$.

$A$ 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 $(x=0, y=-h)$.

$A$ thin ring of mass $2 \ kg$ has a radius of $0.5 \ m$. It is rolling without slipping on a horizontal plane with a velocity of $1 \ m/s$. $A$ small ball of mass $0.1 \ kg$ moving in the opposite direction with a velocity of $20 \ m/s$ hits the ring at a height of $0.75 \ m$ and moves vertically upward with a velocity of $10 \ m/s$ after the collision. Immediately after the collision: