Question

Calculate the radius of gyration of a rod of mass 100 g and length100 cm about an axis passing through its centre of gravity andperpendicular to its length.

Answer 1

uniform rod => density = d = M / L

Take a small length  dx of the rod at a distance x from the center of gravity, which is same as the geometric center.  Its mass  dm = d * dx

  moment of inertia = 2 *   Sum of  dm * x²      for  x = 0 to L/2
         I = 2 * Integral of   d * dx  * x²
           = 2 * d * [ x³/3 ] from x = 0 to L/2
           = d L³ / 12  = M L² / 12

  I = M * R² = M L² /12

 Radius of gyration = R = L/2√3   =  50/√3  cm