$A$ mass $m$ is tied to one end of a spring and whirled in a horizontal circle with constant angular velocity. The elongation in the spring is $1 \ cm$. If the angular speed is doubled,the elongation in the spring is $6 \ cm$. The original length of the spring is (in $cm$)

$A$ disc of mass $0.2 \ kg$ is kept floating in air without falling by vertically firing bullets each of mass $0.05 \ kg$ on the disc at the rate of $10$ bullets per second. If the bullets rebound with the same speed,then the speed of each bullet is (Acceleration due to gravity $g = 10 \ m \ s^{-2}$) (in $m \ s^{-1}$)

$A$ bullet of mass $0.1\, kg$ is fired into a wooden block to pierce through it,but it stops after moving a distance of $50\, cm$ into it. If the velocity of the bullet before hitting the wood is $10\, m/s$ and it slows down with uniform deceleration,then the magnitude of the effective retarding force on the bullet is $'x'\, N$. The value of $'x'$ to the nearest integer is:

In the diagram,$BAC$ is a rigid fixed rough wire and angle $BAC$ is $60^o$. $P$ and $Q$ are two identical rings of mass $m$ connected by a light elastic string of natural length $2a$ and elastic constant $k = \frac{mg}{a}$. If $P$ and $Q$ are in equilibrium when $PA = AQ = 3a$,then the least coefficient of friction between the ring and the wire is $\mu$. Find the value of $\mu + \sqrt{3}$.

Two pulley arrangements shown in the figure are identical. The mass of the rope is negligible. In figure $(a)$,the mass $m$ is lifted by attaching a mass $2m$ to the other end of the rope. In figure $(b)$,$m$ is lifted up by pulling the other end of the rope with a constant downward force $F = 2mg$. The accelerations of $m$ in the two cases are respectively:

$A$ force $F$ is applied on the top of a cube as shown in the figure. The coefficient of friction between the cube and the ground is $\mu$. If $F$ is gradually increased,the cube will topple before sliding. Then the range of $\mu$ is: