A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity $$\omega$$. A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If E be the initial Kinetic energy of the system, then final Kinetic energy will be $$\frac{E}{x}$$. The value of $$x$$ is

A solid sphere of mass $$500 \mathrm{~g}$$ and radius $$5 \mathrm{~cm}$$ is rotated about one of its diameter with angular speed of $$10 ~\mathrm{rad} ~\mathrm{s}^{-1}$$. If the moment of inertia of the sphere about its tangent is $$x \times 10^{-2}$$ times its angular momentum about the diameter. Then the value of $$x$$ will be ___________.

A force of $$-\mathrm{P} \hat{\mathrm{k}}$$ acts on the origin of the coordinate system. The torque about the point $$(2,-3)$$ is $$\mathrm{P}(a \hat{i}+b \hat{j})$$, The ratio of $$\frac{a}{b}$$ is $$\frac{x}{2}$$. The value of $$x$$ is -

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity $$3 \mathrm{~m} / \mathrm{s}$$ (as shown in figure). Maximum height with respect to the initial position covered by it will be __________ cm.