(i) Radius of Gyration

Consider an area A which has moment of Inertia IX with respect to x-axis. Let us imagine this area A to be concerntrated into a thin strip parallel to x-axis, such that it has the same moment of inertia IX.

From definition, $I_x=Ak_x^2 $ where kx is known as the radius of gyration of the area w.r.t. x-axis.

Similarly,

$k_x=\sqrt{\dfrac{I_x}{A}}\ ...\ and\ ...\ k_y=\sqrt{\dfrac{I_y}{A}} $

In general,

$k=\sqrt{\dfrac{I}{A}} $

Therefore, “Radius of Gyration is the distance at which the given area is compressed and kept as a strip of negligible width, such that there is no change in moment of inertia about the given axis”.

Radius of Gyration has units of length and is used in Structural Mechanics in the design of Columns.

(ii) Work-Energy principle

Consider a particle of mass ‘m’ acted upon by a force F and moving from position 1 to 2 along a path which is either rectilinear or curved.

$F_t=ma_t $

$F\cos{\alpha{}}=m.\dfrac{dv}{dt} $

$=m.\dfrac{dv}{ds}.\dfrac{ds}{dt} $

$=m.v.\dfrac{dv}{ds} $

$\therefore{}\left(F\cos{\alpha{}}\right)ds=m.v.dv\ $

Integrating from 1 where s = s1 & v = v1 to 2 where s = s2 & v = v2

$\displaystyle\int_{s_1}^{s_2}F\cos{\alpha{}}.ds=\int_{v_1}^{v_2}mv.dv $

$U_{1-2}=\dfrac{1}{2}mv_2^2-\dfrac{1}{2}mv_1^2 $

Where U1-2 is the work of the force F exerted on the particle during displacement from 1 to 2.

Now, 1/2 mv2 is defined as Kinetic Energy, and denoted as-

$\therefore{}U_{1-2}=T_2-T_1 $

This relation can be extended to a system of forces. The work-energy principle states “For a [article moving under the action of forces, the total work done by these forces is equal to the change in its Kinetic Energy”.