Question

Show that radius of gyration of disc about transverse axis through centre of mass is equal to radius of gyration of a ring about an axis coinciding with its diameter, if disc and ring have same radius.

Answer 1

Hint: The radius of gyration is defined mathematically as the root mean square distance of the object parts from the center of mass or a given axis. We can calculate the radius of gyration if we know the moment of inertia and the total mass of the body.

Complete Step by step solution:
Moment of inertia of anybody can be written as
$I = M{K^2}$
Where
I is the moment of inertia
M is the total mass of the body
K is the radius of gyration

So, the radius of any extended body can be written as
$K = \sqrt {\dfrac{I}{M}} $ ------(1)

Suppose, we have an axis that passes perpendicular to the center of the disc.
The moment of motion of disc about transverse axis is
$I = \dfrac{{M{R^2}}}{2}$

Substituting equation I and M values we get
$\eqalign{
  & \Rightarrow K = \sqrt {\dfrac{{M{r^2}}}{{2M}}} \cr
  & \Rightarrow K = \dfrac{r}{{\sqrt 2 }} \cr
  & \therefore K = \sqrt {\dfrac{I}{M}} \cr} $

So, the radius of gyration is the same for both cases having the same mass.

Additional information:
In simple words, a moment of inertia can be defined as a measurement of resistance to rotational acceleration. It is measured in the unit of kg ${m^2}$.
Inertia is nothing but the property of matter which resists change in its state of motion. And the moment of inertia depends not only on the mass but also the distribution of mass around the axis about which the moment of inertia is to be calculated.

Note:
The radius of gyration is also known as gyroradius, we can also be defined as the radial distance to a point that 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. In other words, the radius of gyration is defined as the radial distance from the rotational axis at which the entire body mass is supposed to be concentrated.