$A$ 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

As shown in the figure,two strings of length $l$ each are attached to a vertical axis $AB$ of length $l$. The strings are $AC$ and $BC$. At point $C$,a point mass $m$ is attached. The mass rotates about the axis with angular velocity $\omega$. The tensions in strings $AC$ and $BC$ are $T_1$ and $T_2$ respectively. Choose the $CORRECT$ alternative:

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 $N$)

If mass,speed,and radius of the circular path of a particle are increased by $100 \%$,then the necessary force required to maintain the circular path will have to be increased by (in $\%$)

When helical gear $M$ turns as shown,gears $I$ & $H$ turn in the following manner. Which of the following is correct? (Assuming no slipping anywhere)

$A$ child of mass $5\,kg$ is going round a merry-go-round that makes $1$ rotation in $3.14\,s$. The radius of the merry-go-round is $2\,m$. The centrifugal force on the child will be $.......\,N$.