During a test of a jet engine,its compressor rotates according to the graph shown. The total number of revolutions completed by the compressor during the test is:

$A$ block of mass $m$ is attached to a pulley disc of equal mass $m$ and radius $r$ by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of $5\, m/s$. Its velocity when the string becomes taut will be

$A$ thin uniform rod of mass $M$ and length $L$ has its moment of inertia $I_1$ about its perpendicular bisector. The rod is bent in the form of a semicircular arc. Now its moment of inertia through the centre of the semicircular arc and perpendicular to its plane is $I_2$. The ratio of $I_1 : I_2$ will be

$A$ wagon of $200\, kg$ is moving on a smooth track with a velocity of $2\, m/s$. $A$ man of $80\, kg$ runs in the wagon with a velocity such that the speed of the centre of mass of the system is zero. Find the relative velocity of the man with respect to the wagon in $m/s$.

The position of an object having mass $0.1 \text{ kg}$ as a function of time $t$ is given as $\vec{r} = (10t^2\hat{i} + 5t^3\hat{j}) \text{ m}$. At $t = 1 \text{ s}$, which of the following statements are correct?
$A$. The linear momentum $\vec{p} = (2\hat{i} + 1.5\hat{j}) \text{ kg} \cdot \text{m/s}$.
$B$. The force acting on the object $\vec{F} = (2\hat{i} + 3\hat{j}) \text{ N}$.
$C$. The angular momentum of the object about its origin $\vec{L} = 15\hat{k} \text{ J} \cdot \text{s}$.
$D$. The torque acting on the object about its origin $\vec{\tau} = 20\hat{k} \text{ N} \cdot \text{m}$.
Choose the correct answer from the options given below:

The moment of inertia of a ring about an axis passing through the centre and perpendicular to its plane is $I$. It is rotating with angular velocity $\omega$. Another identical ring is gently placed on it so that their centres coincide. If both the rings are rotating about the same axis,then the loss in kinetic energy is