$A$ coin placed on a rotating turntable just slips if it is placed at a distance of $4 \ cm$ from the centre. If the angular velocity of the turntable is doubled,it will just slip at a distance of: (in $cm$)

$A$ 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 of mass $m$ is fixed to one end of a light spring having force constant $k$ and unstretched length $\ell$. The other end is fixed. The system is given an angular speed $\omega$ about the fixed end of the spring such that it rotates in a circle in gravity-free space. Then the stretch in the spring is

As shown in the figure, two equal masses hang on either side of a pulley at the same height from the ground. The mass on the right is given a horizontal speed $u$. After some time :-

$A$ 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 ......... $^o$ $(g = 9.8\, m/s^2)$.

$A$ car travelling on a smooth road passes through a curved portion of the road in the form of an arc of a circle of radius $10 \, m$. If the mass of the car is $500 \, kg$,the reaction on the car at the lowest point $P$ where its speed is $20 \, m/s$ is ......... $kN$.