$A$ person with his hands in his pockets is skating on ice at a velocity of $10 \, m/s$ and describes a circle of radius $50 \, m$. What is his inclination with the vertical?

$A$ smooth wire of length $2\pi r$ is bent into a circle and kept in a vertical plane. $A$ bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$ about the vertical diameter $AB$,as shown in the figure,the bead is at rest with respect to the circular ring at position $P$ as shown. Then the value of $\omega^2$ is equal to

$A$ particle of mass $m$ is rotating in a plane in a circular path of radius $r$. Its angular momentum is $L$. The centripetal force acting on the particle is

Three particles of equal mass are tied to a string and rotated in a horizontal plane as shown. What is the ratio of tensions in the three parts of the string?

$A$ hemispherical bowl of radius $R$ is rotated about its axis of symmetry,which is kept vertical,with angular velocity $\omega$. $A$ small block is kept in the bowl. It remains stationary relative to the bowl surface at a position where the radius makes an angle $\theta$ with the vertical. The friction is absent. The value of $\theta$ is

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