Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the velocity of $A$ is $v$ and that of $B$ is $2v$,the velocity of the centre of mass of the system is

$A$ magnetized wire of magnetic moment $M$ and length $L$ is bent into an arc of a circle subtending an angle of $60^{\circ}$ at the center. The new magnetic moment is:

The polar equation of the circle with centre $\left(2, \frac{\pi}{2}\right)$ and radius $3$ units is :

What is the period of small oscillations of the block of mass $m$ if the springs are ideal and pulleys are massless?

$A$ vessel contains hydrogen and nitrogen gases in the ratio $2:3$ by mass. If the temperature of the mixture of the gases is $30^{\circ} C$, then the ratio of the average kinetic energies per molecule of hydrogen and nitrogen gases is $($Molecular mass of hydrogen $=2$ and molecular mass of nitrogen $=28)$

The area (in sq units) bounded by the curves $y^2=4x$ and $x^2=4y$ is