A mass $$m$$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $$m$$ and radius $$R.$$ Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $$m,$$ if the string does not slip on the pulley, is:

A thin horizontal circular disc is rotating about a vertical axis passing through its center. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc.

An object, moving with a speed of 6.25 m/s, is decelerated at a rate given by :
$${{dv} \over {dt}} = - 2.5\sqrt v $$ where v is the instantaneous speed. The time taken by the object, to come to rest, would be :

A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountain is v, the total area around the fountain that gets wet is :