$A$ rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical its lower half separates without exerting any reaction at the breaking point. Then the maximum angle $‘\theta ’$ made by the hinged upper half with the vertical is ......... $^o$.

This question has Statement $1$ and Statement $2$. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement $1$ : When moment of inertia $I$ of a body rotating about an axis with angular speed $\omega $ increases, its angular momentum $L$ is unchanged but the kinetic energy $K$ increases if there is no torque applied on it.
Statement $2$ : $L = I\omega $, kinetic energy of rotation $ = \frac{1}{2}\,I\omega ^2$

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $E_{sphere}/E_{cylinder}$ will be

A tangential force $F$ is applied on a disc of radius $R$, due to which it deflects through an angle $\theta $ from its initial position. The work done by this force would be

Rotational kinetic energy of a given body about an axis is proportional to

A particle performs uniform circular motion with an angular momentum $L.$  If the angular frequency of the particle is doubled and kinetic energy is halved, its angular momentum becomes