What is combined translation and rotation motion?

$A$ disc is rotating with an angular velocity $\omega$. $A$ child sits on it gently. Which of the following quantities will be conserved?

Two discs with moments of inertia $I_1$ and $I_2$ and angular speeds $\omega_1$ and $\omega_2$ are rotating about collinear axes passing through their centers of mass and perpendicular to their planes. If they are joined together to rotate about the same axis,the rotational kinetic energy of the system will be:

An object of mass $m$ is tied to a thin string,which is passed through a hollow tube. The tube is held in one hand and the string in the other. The object is rotated in a circle of radius $R$ with a velocity $v$. If the string is pulled downwards to decrease the radius,which of the following quantities remains conserved?

Which of the following remains constant in a central force field?

If the earth were to suddenly contract to $1/n^{th}$ of its present radius without any change in its mass,the duration (in $hrs.$) of the new day will be nearly