If a person sitting on a rotating stool with his hands outstretched, suddenly lowers his hands, then his

Three uniform similar rods of length $L$ and mass $m$ are joined to form an equilateral triangle. Find the radius of gyration about an axis passing through one corner and perpendicular to the plane of the triangle.

Four rods are arranged in the form of a square. Calculate the moment of inertia about an axis passing through the center and perpendicular to the plane. (Mass $M$ and length $L$ for each rod)

Consider two uniform discs of the same thickness and different radii $R_{1} = R$ and $R_{2} = \alpha R$ made of the same material. If the ratio of their moments of inertia $I_{1}$ and $I_{2}$,respectively,about their axes is $I_{1} : I_{2} = 1 : 16$,then the value of $\alpha$ is:

$A$ solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively $(k_{\text{sph}} : k_{\text{cyl}})$ is $2 : \sqrt{x}$,then the value of $x$ is .............

Moment of inertia of a circular wire of mass $M$ and radius $R$ about its diameter is