A particle describes a horizontal circle of radius 'r' in conical funnel with smooth inner surface with a speed of $0.5$ m/s. The height of the plane of the circle from the vertex of the funnel is (acceleration due to gravity, $g = 10\,\text{m/s}^2$)

A particle at rest starts moving with constant angular acceleration '$\alpha$' in a circular path of radius 'r'. At certain instant, the magnitude of centripetal acceleration is $\left(\dfrac{1}{3}\right)^{rd}$ the tangential acceleration. The relation between linear speed (V) and angular acceleration ($\alpha$) is

A car of mass m is crossing the convex bridge of radius of curvature R with speed V. At the highest point the thrust is ($g$ = gravitational acceleration)

A bucket containing water is revolved in vertical circle of radius r. To prevent the water from falling down, the period of revolution required is ($g$ = gravitational acceleration)