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# 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.

Last updated date: 18th Jun 2024
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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.

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 ${m^2}$.