If the radii of circular paths of two particles of same mass are in the ratio of $1: 2$, then to have a constant centripetal force, the ratio of their speeds should be

$A$ small disc is placed on the top of a smooth hemisphere of radius $R$. The smallest horizontal velocity $V$ that should be imparted to the disc so that it leaves the hemisphere surface immediately is (there is no friction):

$A$ string of length '$L$' fixed at one end carries a mass '$m$' at the other end. The string makes $\frac{3}{\pi}$ r.p.s. around the vertical axis through the fixed end. The tension in the string is (in $mL$)

$A$ stick has its bottom end attached to a wall by a pivot and is held up by a massless string attached to its other end. Which of the following scenarios has the smallest tension in the string? (Length of stick is same in all scenarios)

$A$ boy on a cycle pedals around a circle of $20\, m$ radius at a speed of $20\, m/s$. The combined mass of the boy and the cycle is $90\, kg$. The angle that the cycle makes with the vertical so that it may not fall is ......... $^o$ $(g = 9.8\, m/s^2)$.

$A$ proton of mass $1.6 \times 10^{-27} \, kg$ moves in a circular orbit of radius $0.10 \, m$ under a centripetal force of $4 \times 10^{-13} \, N$. The frequency of revolution of the proton is about: