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.

The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is $$\frac{1}{x} \mathrm{MR}^{2}$$, where $$\mathrm{R}$$ is the radius and $$M$$ is the mass of the semicircular ring. The value of $$x$$ will be __________.