Physics System of Particles and Rotational Motion

Kindly go through the solution 

Radius of gyration of a solid surface,

${\mathrm{K}}_{\mathrm{S}}=\sqrt[]{\frac{2}{5}\mathrm{R}}$

Radius of gyration of a hollow surface,

${\mathrm{K}}_{\mathrm{H}}=\sqrt[]{\frac{2}{3}}\mathrm{R}$

$\Rightarrow \frac{{\mathrm{K}}_{\mathrm{S}}}{{\mathrm{K}}_{\mathrm{H}}}=\sqrt[]{\frac{3}{5}}=\frac{\sqrt[]{3}}{\sqrt[]{5}}$

Yes, an object moving in a straight path can also possess angular momentum with respect to a referenc epoint which doesn't lie on it's path. In such cases. the referenc epoint may lie perpendicular distance from the axis. If an object is moving in a straight line (not through the reference point) the perpendicular distance $r_\perp$ (l = rp) will be non-zero.

Yes, how much time these planets take in completing an orbit around the sun hugely depends on their distance from the sun as well as their mass. Since these planets move in a circular orbit, the term angular momentum is used in such cases for computing relevant information.

Angular momentum also depends on how the mass is spread over the body and not just mass only. If you closely look at the formula of angular momentum:

L=r*p

You will observe that it is a cross product of two vector quantities. Hence, the formula won't be practical without direction being involved. 

This is because the terms used in rotational motion are analogous to the terms used in linear motion and can interchange each other with variation in the topic since their meaning is exactly the same. For example, what you call force in linear motion can be called torque in terms of rotational motion. Just replace the linear terms with angular terms and you are done with the major process.

This is because using radians will simplify the procedure of solving various numerical problems and complex mathematical equations. They help consider the terms such as arc length, centre, radius etc. directly into account which are connected to a circular path which eases our task of computing the formulas.

This is the type of acceleration due to which a circular moment of the object is possible. This force constantly directs the motion of the body towards the centre of gravitation. Without this motion, the object will move away in a straight line. It ensures that the object stays in a circular path by changing the direction of object's velocity towards the centre.

Yes. Angular Velocity is a vector quantity i.e. it has directions. The direction of rotation can influence the sign of angular velocity's value. If the movement of the object is anticlockwise, the sign will be positive. Similarly, if the movement of the object is clockwise, the sign will be negative.