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

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

A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is $20\, m.$ The speed of the car at the top of the hill is between

At time $t=0$, a disk of radius $1 m$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha=\frac{2}{3} rad s ^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time $t=\sqrt{\pi} s$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in $m$ ) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$. The value of $x$ is. . . . . . .[Take $g=10 m s ^{-2}$.]

A boy is sitting on the horizontal platform of a joy wheel at a distance of $5 \,m$ from the center. The wheel begins to rotate and when the angular speed exceeds $1 \,rad / s$, the boy just slips. The coefficient of friction between the boy and the wheel is $\left(g=10 \,m / s ^2\right)$

A car is moving on a horizontal curved road with radius $50\,m$. The approximate maximum speed of car will be $............\,ms^{-1}$, if friction between tyres and road is $0.34.\left[\right.$ Take $\left.g =10 ms ^{-2}\right]$