question_answer

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is

A) \[1:\sqrt{2}\]

B) l : 3  

C) 2: l   

D) \[\sqrt{5}:\sqrt{6}\]

Correct Answer: D

Solution :

[d]        \[{{I}_{{{y}_{1}}}}=\frac{M{{R}^{2}}}{4}\] \[\therefore \,\,I{{'}_{{{y}_{1}}}}=\frac{M{{R}^{2}}}{4}+M{{R}^{2}}=\frac{5}{4}M{{R}^{2}}\] . \[{{I}_{{{y}_{2}}}}=\frac{M{{R}^{2}}}{2}\] \[\therefore \,\,I{{'}_{{{y}_{2}}}}=\frac{M{{R}^{2}}}{2}+M{{R}^{2}}=\frac{3}{2}M{{R}^{2}}\] \[I{{'}_{{{y}_{1}}}}=MK_{1}^{2},\,\,I{{'}_{{{y}_{2}}}}=MK_{2}^{2}\] \[\therefore \,\,\,\frac{K_{1}^{2}}{K_{2}^{2}}=\frac{I{{'}_{{{y}_{1}}}}}{I{{'}_{{{y}_{2}}}}}\Rightarrow {{K}_{1}}:{{K}_{2}}=\sqrt{5}:\sqrt{6}\]