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 $\%$)

$A$ particle is released on a vertical smooth semicircular track from point $X$ so that $OX$ makes an angle $\theta$ from the vertical (see figure). The normal reaction of the track on the particle vanishes at point $Y$ where $OY$ makes an angle $\phi$ with the horizontal. Then

Assume a proton is rotating along a circular path of radius $1 \,m$ under a centrifugal force of $4 \times 10^{-12} \,N$. If the mass of the proton is $1.6 \times 10^{-27} \,kg$, then its angular velocity of rotation is

$A$ block of $200\, g$ mass moves with a uniform speed in a horizontal circular groove,with vertical side walls of radius $20\, cm$. If the block takes $40\, s$ to complete one round,the normal force by the side walls of the groove is

An aircraft executes a horizontal loop at a speed of $720 \; km/h$ with its wings banked at $15^{\circ}$. What is the radius of the loop in $km$?

If the string of a conical pendulum makes an angle $\theta$ with the horizontal,then the square of its time period is proportional to