The position vector of a particle is given by $\vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k}$ units. $A$ force $\vec{F} = p\hat{i} + 3\hat{j} + 6\hat{k}$ units acts on it. For what value of $p$ is the angular momentum conserved?

$A$ solid sphere is rotating in free space. If the radius of the sphere is increased while keeping its mass constant,which of the following quantities will remain unaffected?

$A$ particle is moving in a circular orbit of radius $r_1$ with an angular velocity $\omega_1$. It jumps to another circular orbit of radius $r_2$ and attains an angular velocity $\omega_2$. If $r_2 = 0.5\, r_1$ and assuming that no external torque is applied to the system,then the angular velocity $\omega_2$ is given by:

$A$ circular plate is rotating in a horizontal plane about an axis passing through its center and perpendicular to the plate with an angular velocity $\omega$. $A$ person sits at the center holding two dumbbells. When he stretches out his hands,the moment of inertia of the system becomes triple. If $E$ is the initial kinetic energy of the system,then the final kinetic energy will be $\frac{E}{x}$. The value of $x$ is $....$

$A$ circular disk of moment of inertia $I_t$ is rotating in a horizontal plane,about its symmetry axis,with a constant angular speed $\omega_i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially,the second disk has zero angular speed. Eventually,both the disks rotate with a constant angular speed $\omega_f$. The energy lost by the initially rotating disk to friction is

The Sun rotates around its centre once in $27$ days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence (in $days$)? Assume the Sun to be a sphere of uniform density.