$A$ ball of mass $0.5 \ kg$ is attached to a string of length $50 \ cm$. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is $400 \ N$. The maximum possible value of angular velocity of the ball in $rad/s$ is:

$A$ 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 $P$?

$A$ ball of mass $m = 0.5 \ kg$ is attached to the end of a string having length $L = 0.5 \ m$. The ball is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is $324 \ N$. The maximum possible value of angular velocity of the ball (in $rad/s$) is

$A$ vehicle of mass $200\,kg$ is moving along a levelled curved road of radius $70\,m$ with an angular velocity of $0.2\,rad/s$. The centripetal force acting on the vehicle is $.........\,N$.

$A$ train is moving towards north. At one place it turns towards north-east,here we observe 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