A coin is placed on a disc. The coefficient of friction between the coin and the disc is $$\mu$$. If the distance of the coin from the center of the disc is $$r$$, the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is :

A stone of mass $$900 \mathrm{~g}$$ is tied to a string and moved in a vertical circle of radius $$1 \mathrm{~m}$$ making $$10 \mathrm{~rpm}$$. The tension in the string, when the stone is at the lowest point is (if $$\pi^2=9.8$$ and $$g=9.8 \mathrm{~m} / \mathrm{s}^2$$) :

If the radius of curvature of the path of two particles of same mass are in the ratio $$3: 4$$, then in order to have constant centripetal force, their velocities will be in the ratio of :

A train is moving with a speed of $$12 \mathrm{~m} / \mathrm{s}$$ on rails which are $$1.5 \mathrm{~m}$$ apart. To negotiate a curve radius $$400 \mathrm{~m}$$, the height by which the outer rail should be raised with respect to the inner rail is (Given, $$g=10 \mathrm{~m} / \mathrm{s}^2)$$ :