Question

radius of gyration of a disc of mass 5kg, about its a transverse axis passing through its centre is 14.14cm.find radius of gyration about its diameter and hence calculate mi about the diameter​
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Answer 1

Given:

Radius of gyration of a disc of mass 5 kg, about it's transverse axis passing through its centre is 14.14 cm.

To Find:

What is radius of gyration about its diameter and also calculate moment of inertia about the diameter​.

Solution:

$\mathbf{\textrm{Radius of gyration about polar axis} = K_{p}=14.14 \ \ \ cm=0.1414 \ \ m}$

$\mathbf{\textrm{Radius of gyration about diameter axis} = K_{d}=Unknown}$

$\mathbf{\textrm{Moment of inertia about polar axis} = I_{p}=M\times K_{p}^{2}=5\times 0.1414^{2}}$

$\mathbf{\textrm{Moment of inertia about diameter} = I_{d}=M\times K_{d}^{2}}$

We know relation between moment of inertia about polar or transverse axis and moment of inertia about diameter axis, which are given below:

$\mathbf{2\times I_{d}=I_{p}}$

So:

$\mathbf{I_{d}=\dfrac{I_{p}}{2}=\dfrac{5\times 0.1414^{2}}{2}}$

$\mathbf{I_{d}=0.04998 \ \ \ kg-m^{2}}$

L.H.S of above equation can be written as

$\mathbf{M\times K_{d}^{2}=0.04998 \ \ \ kg-m^{2}}$

$\mathbf{5\times K_{d}^{2}=0.04998 \ \ \ kg-m^{2}}$

On simplify:

$\mathbf{K_{d}= 0.0999 \ \ m=9.99 \ \ cm}$

Means moment of inertia about diameter axis is 0.04998 kg.m² and radius of gyration about diameter axis is 9.99 cm.