When a disc is rotating with angular velocity $\omega$,a particle situated at a distance of $4 \ cm$ just begins to slip. If the angular velocity is doubled,at what distance will the particle start to slip (in $cm$)?

If the radius of curvature of the path of two particles of same masses are in the ratio $1 : 2$,then in order to have constant centripetal force,their velocity,should be in the ratio of

If a stone of mass $0.5 \ kg$ tied to one end of a wire is whirled in a circular path of radius $2 \ m$ with a speed $40 \ rev/min$ in a horizontal plane,then the tension in the wire is nearly (in $N$)

$A$ cyclist riding a bicycle at a speed of $14\sqrt{3} \text{ m/s}$ takes a turn around a circular road of radius $20\sqrt{3} \text{ m}$ without skidding. Given $g = 9.8 \text{ m/s}^2$,what is his inclination to the vertical in degrees?

$A$ 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

$A$ dancer moves counterclockwise at a constant speed around the path shown below. The path is such that the lengths of its segments,$PQ, QR, RS$,and $SP$,are equal. Arcs $QR$ and $SP$ are semicircles. Which of the following best represents the magnitude of the dancer's acceleration as a function of time $t$ during one trip around the path,beginning at point $P$?