Identify the increasing order of the angular velocities of the following:
$1$. Earth rotating about its own axis
$2$. Hour hand of a clock
$3$. Second hand of a clock
$4$. Flywheel of radius $2 \ m$ making $300 \ rpm$

$A$ wheel initially at rest,begins to rotate about its axis with constant angular acceleration. If it rotates through an angle $\theta_1$ in first $2 \ s$ and a further angle $\theta_2$ in the next $2 \ s$,the ratio $\theta_1 : \theta_2$ is

$A$ particle moves from point $P_1$ to $P_2$ along a circular path with constant speed $v$. If the angle subtended by the arc $P_1P_2$ at the center is $90^{\circ}$,what is the change in velocity?

Three particles $A, B$ and $C$ move in a circle of radius $r = \frac{1}{\pi} \, m$ in the anticlockwise direction with speeds $1 \, m/s$,$2.5 \, m/s$ and $2 \, m/s$ respectively. The initial positions of $A, B$ and $C$ are as shown in the figure. The ratio of the distance travelled by $B$ and $C$ by the instant $A, B$ and $C$ meet for the first time is

$A$ particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c = k^2rt^2$,where $k$ is a constant. What is the power delivered to the particle by the force acting on it?

$A$ particle is revolving in a circle of radius $2 \ m$ with angular velocity $\omega = t^2 - 4t + 8 \ rad/s$. The time when the speed of the particle becomes $8 \ m/s$ is ......... $\sec$.