$A$ ring of mass $M$ and radius $R$ sliding with a velocity $v_0$ suddenly enters a rough surface where the coefficient of friction is $\mu$,as shown in the figure. Choose the correct statement$(s)$.

The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation:
$\vec{r}(t) = \alpha t^3 \hat{i} + \beta t^2 \hat{j}$
where $\alpha = 10/3 \ m \ s^{-3}$,$\beta = 5 \ m \ s^{-2}$ and $m = 0.1 \ kg$. At $t = 1 \ s$,which of the following statement$(s)$ is(are) true about the particle?
$(A)$ The velocity $\vec{v}$ is given by $\vec{v} = (10 \hat{i} + 10 \hat{j}) \ m \ s^{-1}$
$(B)$ The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L} = -(5/3) \hat{k} \ N \ m \ s$
$(C)$ The force $\vec{F}$ is given by $\vec{F} = (2 \hat{i} + 1 \hat{j}) \ N$
$(D)$ The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau} = -(20/3) \hat{k} \ N \ m$

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?

$A$ flywheel is used with an engine because it

The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. Which graph represents the variation of $I$ with respect to $R$,keeping $M$ constant?

$A$ uniform rod of mass $M$ and length $L$ is pivoted at one end and is free to rotate in a vertical plane. The rod is released from a horizontal position. When the rod becomes vertical,the reaction force at the pivot is: