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?

$A$ particle of mass $m$ is moving along the side of a square of side '$a$',with a uniform speed $v$ in the $x-y$ plane as shown in the figure. Which of the following statements is false for the angular momentum $\vec L$ about the origin?

$A$ disc and a ring of the same mass are rolling. If their kinetic energies are equal,then the ratio of their velocities will be:

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery and sticks to it. The loss in kinetic energy due to the collision is ....... $J$.

The graph between angular momentum $L$ and angular velocity $\omega$ is:

Two thin circular discs of mass $m$ and $4m$,having radii of $a$ and $2a$,respectively,are rigidly fixed by a massless,rigid rod of length $l=\sqrt{24}a$ through their centers. This assembly is laid on a firm and flat surface,and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point $O$ is $\vec{L}$ (see the figure). Which of the following statement$(s)$ is(are) true?
$(A)$ The center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega/5$
$(B)$ The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81ma^2\omega$
$(C)$ The magnitude of angular momentum of the assembly about its center of mass is $17ma^2\omega/2$
$(D)$ The magnitude of the $z$-component of $\vec{L}$ is $55ma^2\omega$