$A$ mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed centre at an angular velocity $\omega_0$. If the length of the string and angular velocity are doubled,the tension in the string which was initially $T_0$ is now

$A$ particle of mass '$m$' is rotating along a circular path of radius '$r$' having angular momentum '$L$'. The centripetal force acting on the particle is given by

Two particles of mass $2m$ and $m$ are attached to a light string as shown. The complete system is rotated in a horizontal circle with a constant angular velocity $2\omega$ about an axis passing through point $O$ and perpendicular to the plane of the circle. Find the ratio $T_{OA} / T_{AB}$,where $T_{OA}$ and $T_{AB}$ are the tensions in strings $OA$ and $AB$ respectively.

$A$ mass $m$ is attached to one end of a spring with spring constant $k$ and natural length $l$. The system is rotated in a horizontal plane with an angular velocity $\omega$. What is the extension in the length of the spring?

An aircraft executes a horizontal loop of radius $9 \ km$ at a constant speed of $540 \ kmh^{-1}$. The wings of the aircraft are banked at an angle of (Acceleration due to gravity $g = 10 \ ms^{-2}$)

$A$ stone of mass $2 \,kg$ is tied at one end of a string of length $2 \,m$ and is whirled in a horizontal circle. If the string can withstand a maximum tension of $64 \,N$, then the permissible maximum number of rotations per minute is