$A$ particle is moving in a uniform circular motion with angular momentum $L$. If the frequency of motion is doubled and its kinetic energy is halved,the new angular momentum will be ...

$A$ uniform rod $AB$ of length $1 \ m$ and mass $4 \ kg$ is sliding along two mutually perpendicular frictionless walls $OX$ and $OY$. The velocity of the two ends of the rod $A$ and $B$ are $3 \ m/s$ and $4 \ m/s$ respectively, as shown in the figure. Which of the following statement$(s)$ is/are correct?

$A$ disc having velocity $v_0$ and angular speed $\omega_0$ in the anticlockwise direction is placed on a rough plank. Initially,the plank is at rest. (Assume the length of the plank is very large). Choose the $INCORRECT$ option.

$A$ 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?

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. $(a)$ What is the angular speed of the two-disc system? $(b)$ Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$.

$A$ wheel of moment of inertia $2 \ kg \ m^2$ is rotating about an axis passing through its centre and perpendicular to its plane at a speed of $60 \ rad \ s^{-1}$. Due to friction, it comes to rest in $5$ minutes. The angular momentum of the wheel three minutes before it stops rotating is: