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)

A particle performing U.C.M. of radius $\dfrac{\pi}{2}$ m makes 'x' revolutions in time t. Its tangential velocity is