$A$ particle moves with constant angular velocity in a circular path of a certain radius and is acted upon by a certain centripetal force $F$. If the angular velocity is doubled,keeping the radius the same,the new force will be

$A$ particle rotates in a horizontal circle of radius $r$ in a conical funnel with speed $v$. The inner surface of the funnel is smooth. The height $h$ of the plane of the circle from the vertex of the funnel is (where $g$ is the acceleration due to gravity):

$A$ conical pendulum of length $1\,m$ makes an angle $\theta = 45^\circ$ with respect to the $Z$-axis and moves in a circle in the $XY$ plane. The radius of the circle is $0.4\,m$ and its centre is vertically below $O$. The speed of the pendulum,in its circular path,will be ..... $m/s$ (Take $g = 10\,ms^{-2}$)

$A$ particle describes a horizontal circle of radius $r$ on the smooth surface of an inverted cone as shown. The height of the plane of the circle above the vertex is $h$. The speed of the particle should be

$A$ string of length $0.1\,m$ cannot bear a tension more than $100\,N$. It is tied to a body of mass $100\,g$ and rotated in a horizontal circle. The maximum angular velocity can be .......... $rad/s$.

Two particles of the same mass are moving in circular orbits because of a force given by $F(r) = -\frac{16}{r} - r^3$. The first particle is at a distance $r = 1$,and the second is at $r = 4$. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to: