$A$ mass $M = 40 \ kg$ is fixed at the very edge of a long plank of mass $80 \ kg$ and length $1 \ m$ which is pivoted such that it is in equilibrium. How far (approx.) from the pivot should a mass of $100 \ kg$ be attached so that the plank starts rotating with an angular acceleration of $1 \ rad/s^2$?

$A$ uniform circular disc of mass $1.5 \ kg$ and radius $0.5 \ m$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5 \ N$ are applied simultaneously along the three sides of an equilateral triangle $XYZ$ with its vertices on the perimeter of the disc (see figure). One second after applying the forces,the angular speed of the disc in $\text{rad } s^{-1}$ is:

Which one of the following four graphs best depicts the variation with $x$ of the moment of inertia $I$ of a uniform triangular lamina about an axis parallel to its base at a distance $x$ from it?

The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. If the mass is kept constant,what will be the nature of the graph between $I$ and $R$?

The position of an object having mass $0.1 \text{ kg}$ as a function of time $t$ is given as $\vec{r} = (10t^2\hat{i} + 5t^3\hat{j}) \text{ m}$. At $t = 1 \text{ s}$, which of the following statements are correct?
$A$. The linear momentum $\vec{p} = (2\hat{i} + 1.5\hat{j}) \text{ kg} \cdot \text{m/s}$.
$B$. The force acting on the object $\vec{F} = (2\hat{i} + 3\hat{j}) \text{ N}$.
$C$. The angular momentum of the object about its origin $\vec{L} = 15\hat{k} \text{ J} \cdot \text{s}$.
$D$. The torque acting on the object about its origin $\vec{\tau} = 20\hat{k} \text{ N} \cdot \text{m}$.
Choose the correct answer from the options given below:

$A$ circular hoop of mass $m$ and radius $R$ rests flat on a horizontal frictionless surface. $A$ bullet,also of mass $m$ and moving with a velocity $v$,strikes the hoop and gets embedded in it. The thickness of the hoop is much smaller than $R$. The angular velocity with which the system rotates after the bullet strikes the hoop is