A disc of mass $$1 \mathrm{~kg}$$ and radius $$\mathrm{R}$$ is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be $$4 \sqrt{\frac{x}{3 R}} \,\operatorname{rad}{s}^{-1}$$ where $$x=$$ ____________. $$\left(g=10 \mathrm{~ms}^{-2}\right)$$
Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 3 m each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $$\sqrt x$$ m. The value of x is ____________.
Four particles with a mass of 1 kg, 2 kg, 3 kg and 4 kg are situated at the corners of a square with side 1 m (as shown in the figure). The moment of inertia of the system, about an axis passing through the point O and perpendicular to the plane of the square, is ______________ kg m2.
The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is I1. The same rod is bent into a ring and its moment of inertia about a diameter is I2. If $${{{I_1}} \over {{I_2}}}$$ is $${{x{\pi ^2}} \over 3}$$, then the value of x will be ____________.