The minimum horizontal speed with which a body must be projected, so that it goes around a smooth vertical circular track of radius $4 \,m$ is $(g=9.8 \,ms^{-2})$. (in $\,ms^{-1}$)

$A$ particle originally at rest at the highest point of a smooth vertical circle is slightly displaced. It will leave the circle at a vertical distance $h$ below the highest point such that

$A$ light rigid wire of length $1 \,m$ is attached to a ball of mass $500 \,g$ at one end. The other end of the wire is fixed, so that the wire can rotate freely in the vertical plane about its fixed end. At the lowest point of the circular motion, the ball is given a horizontal velocity of $6 \,m/s$. Determine the radial component of the acceleration of the ball when this rigid wire makes an angle of $60^{\circ}$ with the upward vertical. (Take $g = 10 \,m/s^2$) (in $\,m/s^2$)

$A$ ball of mass $0.1 \ kg$ is suspended by a string. It is displaced through an angle of $60^\circ$ and released. When the ball passes through the mean position,the tension in the string is ........ $N$.

What will be the minimum speed of the roller-coaster so that the passenger at the top, when becomes upside down, does not fall out (in $m \ s^{-1}$)? Consider the acceleration due to gravity as $10 \ m \ s^{-2}$, and the radius of curvature of the roller coaster is $10 \ m$.

$A$ particle is tied to a $20 \ cm$ long string. It performs circular motion in a vertical plane. What is the angular speed of the particle when the tension in the string at the top is zero? (Take $g = 9.8 \ m/s^2$)