Question

What is the radius of gyration?

Answer 1

Hint: There are 2 ways of compressive stress failure in columns – Crushing and Buckling. Crushing occurs for shorter columns, also called struts and Buckling occurs in longer columns. Even though you may think that their classification of long and short columns is subjective, it depends on a number known as Slenderness ratio and this slenderness ratio depends on a quantity called radius of gyration.

Complete step-by-step answer:
Whenever a body is set into motion, it is observed that the body will not respond quickly to the rotation. This is due to the fact there is something called inertia which indicates the inability to adapt to the quick change in the state of the body. When this inertia is associated with rotation, we call it as moment of inertia.
The moment of inertia basically, gives us the distribution of mass of a body around its axis of rotation and it is the quantity that determines the torque required to produce an angular acceleration in the rotating body.
If a body consists of n particles of mass m each at distance of ${r_1},{r_2}..{r_n}$ respectively, then
Moment of inertia,
$
  I = m{r_1}^2 + m{r_2}^2 + m{r_3}^2 + .. + m{r_n}^2 \\
   \to I = m\left( {{r_1}^2 + {r_2}^2 + {r_3}^2 + .. + {r_n}^2} \right) \\
 $
This whole term can be replaced as,
${k^2} = \left( {{r_1}^2 + {r_2}^2 + {r_3}^2 + .. + {r_n}^2} \right)$
where k is called the radius of gyration.
Thus, radius of gyration can be defined as the Radius of gyration of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated.

Note: In the hint, We mentioned the term slenderness ratio. This is how it is related to radius of gyration.
Slenderness ratio = $\dfrac{l}{k}$
where $l$= effective length of the column
and $k$= radius of gyration.
The radius of gyration, $k = \sqrt {\dfrac{I}{A}} $ where $I$is the least moment of inertia of the cross-section of the column and A is the area of cross-section.