The graph between $\log_e L$ and $\log_e P$ will be (where $L$ is angular momentum and $P$ is linear momentum):

State whether the following statements are true or false:
$(1)$ Torque produces angular velocity in an object.
$(2)$ For the rotational motion of a rigid body,the linear variables of all its particles are the same.

Find the increase in the moment of inertia $I$ of a uniform rod (coefficient of linear expansion $\alpha$) about its perpendicular bisector when its temperature is slightly increased by $\Delta T$.

In the following problem,indicate the correct direction of the friction force acting on a cylinder of mass $M$ and radius $R$,which is pulled on a rough surface by a constant horizontal force $F$ applied at its center. The friction force can be represented by which of the following diagrams?

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 $\omega_{1} \neq \omega_{2}$.

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