A circular disc reaches from top to bottom of an inclined plane of length $$l$$. When it slips down the plane, if takes $$t \mathrm{~s}$$. When it rolls down the plane then it takes $$\left(\frac{\alpha}{2}\right)^{1 / 2} t \mathrm{~s}$$, where $$\alpha$$ is _________.

A string is wrapped around the rim of a wheel of moment of inertia $$0.40 \mathrm{~kgm}^2$$ and radius $$10 \mathrm{~cm}$$. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $$40 \mathrm{~N}$$. The angular velocity of the wheel after $$10 \mathrm{~s}$$ is $$x \mathrm{~rad} / \mathrm{s}$$, where $$x$$ is __________.

A circular table is rotating with an angular velocity of $$\omega \mathrm{~rad} / \mathrm{s}$$ about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of $$1 \mathrm{~m}$$ on the groove. All the surfaces are smooth. If the radius of the table is $$3 \mathrm{~m}$$, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $$x \sqrt{2} \omega \mathrm{~m} / \mathrm{s}$$, where the value of $$x$$ is _________.

Three balls of masses $$2 \mathrm{~kg}, 4 \mathrm{~kg}$$ and $$6 \mathrm{~kg}$$ respectively are arranged at centre of the edges of an equilateral triangle of side $$2 \mathrm{~m}$$. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ________ $$\mathrm{kg} \mathrm{~m}^2$$.