$A$ conical pendulum is moving in a circle with angular velocity $\omega$ as shown. If the tension in the string is $T$,which of the following equations is correct?

$A$ cage revolves around a vertical circle of radius $R$ with constant linear speed $\sqrt{gR}$. The cage is connected to the revolving arm in such a manner that a boy of mass $m$ remains always vertical while standing on a weighing machine kept inside the cage. It is found that

Two masses $M$ and $m$ are attached to a vertical axis by weightless threads of combined length $l$. They are set in rotational motion in a horizontal plane about this axis with constant angular velocity $\omega$. If the tensions in the threads are the same during motion,the distance of $M$ from the axis is

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.

$A$ cyclist goes round a circular path of circumference $34.3 \ m$ in $\sqrt{22} \ s$. The angle made by him with the vertical will be ....... $^o$.

Find the centripetal force acting on a coin weighing $0.1 \,kg$ placed $0.1 \,m$ from the centre on a gramophone disc rotating at $600 \,rpm$.